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AN APPLICATION OF THE MAXIMUM PRINCIPLE TODESCRIBE THE LAYER BEHAVIOR OF LARGE SOLUTIONS
AND RELATED PROBLEMS
JORGE GARCIA-MELIAN, JULIO D. ROSSI, AND JOSE C. SABINA DE LIS 1
Abstract. This work is devoted to the analysis of the asymptotic behavior of positivesolutions to some problems of variable exponent reaction-diffusion equations, when theboundary condition goes to infinity (large solutions). Specifically, we deal with the
equations âu = up(x), âu = âm(x)u + a(x)up(x) where a(x) â„ a0 > 0, p(x) â„ 1 in
Ω, and âu = ep(x) where p(x) â„ 0 in Ω. In the first two cases p is allowed to takethe value 1 in a whole subdomain Ωc â Ω, while in the last case p can vanish in awhole subdomain Ωc â Ω. Special emphasis is put in the layer behavior of solutionson the interphase Îi := âΩc ⩠Ω. A similar study of the development of singularities inthe solutions of several logistic equations is also performed. For example, we considerââu = λm(x)uâa(x)up(x) in Ω, u = 0 on âΩ, being a(x) and p(x) as in the first problem.Positive solutions are shown to exist only when the parameter λ lies in certain intervals:bifurcation from zero and from infinity arises when λ approaches the boundary of thoseintervals. Such bifurcations together with the associated limit profiles are analyzed indetail. For the study of the layer behavior of solutions the introduction of a suitablevariant of the well-known maximum principle is crucial.
1. Introduction
Let L be a second order elliptic operator without zero order term
Lu = âNâ
i,j=1
aij(x)â2u
âxiâxj+
Nâ
i=1
bi(x)âu
âxi
with, say, bounded measurable coefficients aij , bi defined in a bounded domain Ω â RN ,which satisfies a strict ellipticity condition under the form
(1.1)Nâ
i,j=1
aij(x)ΟiΟj ℠”|Ο|2,
for all x â Ω, Ο â RN and a fixed positive ” (see [12]). The well-known maximum principleâor more properly the âweakâ maximum principleâ states that, provided a function u âC2(Ω) â© C(Ω) satisfies
(1.2) Lu â„ 0
Date: November 18, 2009.2000 Mathematics Subject Classification. 35B50, 35J65, 35B32.Key words and phrases. Maximum principle, sub and supersolutions, bifurcation, boundary blow-up,
large solutions.1 Corresponding author.
1
2 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
in Ω, then its infimum in Ω coincides with the corresponding infimum when restricted toâΩ. That is,
(1.3) infΩ
u = infâΩ
u.
Alternatively, relation (1.3) holds after changing infimum by supremum if the sign ininequality (1.2) is reversed (see [12] and the classical reference on this subject, [20]).
Unfortunately, and as it is also well-known, this nice property may fail when the operatorL is complemented with a zero order term, i.e.,
(L+ c) u = Lu + cu,
where c â Lâ(Ω). However, useful weak versions of the maximum principle still survivefor the perturbed operator L + c and a precise characterization of these variants can begiven. Namely, those expressed in terms of the principal Dirichlet eigenvalue λ1 of L + cin Ω (we will review all these features in Section 2).
In this work we provide a sort of weak alternative to the maximum principle (1.2)â(1.3).Our main objective will be then to employ such result to study different layer behaviorsassociated to the formation of âlargeâ solutions, and to analyze the developing of âinfiniteplateausâ in the solutions of certain reaction-diffusion problems (see [1], [15], and specially[4], [21] for recent surveys on large solutions).
To be more precise, the simple version of the maximum principle which is enough foraddressing the kind of applications we have in mind here is the following.
Theorem 1. Let
(L+ c)u = âNâ
i,j=1
aij(x)â2u
âxiâxj+
Nâ
i=1
bi(x)âu
âxi+ c(x)u
be a strictly elliptic operator, that is L verifies (1.1), with coefficients aij , bi, c â Cα(Ω) de-fined in a C2,α bounded domain Ω â RN . Assume in addition that its principal (Dirichlet)eigenvalue satisfies
λ1 = λ1(L+ c) > 0.
Then,i) The boundary value problem,
(L+ c) Ï(x) = 0 x â Ω
Ï(x) = 1 x â âΩ,
has a unique positive solution Ï â C2,α(Ω).ii) (Maximum principle, restricted version) Suppose that u â C2(Ω) â© C(Ω) satisfies
(L+ c)u â„ 0,
in Ω. Then
(1.4) u(x) â„ infâΩ
uÏ(x),
for all x â Ω.The symmetric statement also holds: if (L+c)u †0 then u(x) †supâΩ uÏ(x).
Remarks 1.a) A definition of the principal eigenvalue λ1(L+ c) of the operator L+ c in the domain Ωis given in Section 2.
MAXIMUM PRINCIPLE 3
b) If one deals with strong solutions instead of classical ones, the smoothness of bothcoefficients and solutions in Theorem 1 can be relaxed (see Remak 4-c)).
As for the applications of this maximum principle we first deal with the limit behavioras n ââ of the solution u = un to the semilinear problem
(1.5)
âu(x) = f(x, u(x)) x â Ωu(x) = n x â âΩ.
Assume that Ω â RN is bounded and C2,α, f â Cα(Ω Ă R) is non decreasing in u anddiverges to infinity as u â â with a suitable strength. Namely, f(x, u) â„ h(u) beingh â C(R) an increasing function such that the âKeller-Ossermanâsâ condition is fulfilled
(1.6)â« â
a
(â« s
0h(u) du
)â1/2
ds < â,
for a certain large a > 0. Then the solution un to (1.5) converges in C2(Ω) to a so-calledâlargeâ solution u, that is, a solution to the boundary blow-up problem
(1.7)
âu = f(x, u) x â Ωu = â x â âΩ.
In fact, references [15], [19] introduce a first systematic study of (1.7) under this approach.Since its appearance in different fields as Riemannian geometry or population dynamics,
large solutions have deserved much interest in the recent pdeâs literature (see the compre-hensive reviews in [21] and [4]). In [7], [9] and [10] it is discussed âamong other aspectsâthe effect of losing condition (1.6) on a whole subdomain Ωc â Ω. This analysis wasessentially performed for the two kind of nonlinearities: f(x, u) = up(x), f(x, u) = ep(x)u.
To summarize the relevant issues of those papers that have implications in the presentwork and to describe the new features here enclosed, it is convenient to introduce a sim-plifying hypothesis on the structure of the critical subdomain Ωc â Ω.
In the different problems we are analyzing, it will be understood by âcriticalâ the failurein Ωc of a certain structural condition which is essential for the existence of solutions(for example, the Keller-Ossermanâs condition in the case of large solutions). It willbe supposed that Ωc and Ω are bounded and C2,α domains such that Îb := âΩc â© âΩ, ifnonempty, consists of the union of components of âΩ. Equivalently, Îi := âΩcâ©Î© is closed,and thus is separated away from âΩ. These conditions on Ωc together with Ωc 6= Ω, willbe referred to as hypothesis (H) in the sequel. At this early stage we anticipate that muchof the contributions of the present research is concerned with the behavior of solutions inthe inner boundary Îi of the critical region Ωc.
Some properties of the power nonlinearity are listed in the next statement, which isconsequence of the analysis in [9] (see also [7] for a related logistic type nonlinearity).
Theorem 2 ([9]). Consider the problem
(1.8)
âu(x) = u(x)p(x) x â Ωu(x) = n x â âΩ.
where p â Cα(Ω) is positive, Ωc â Ω are bounded C2,α domains, Ωc satisfies (H) andwhere
Ωc = x â Ω : p(x) †1.Then the following properties hold true.
4 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
i) Problem (1.8) admits for every n â N a unique solution un â C2,α(Ω).ii) If Ωc â Ω then un converges in C2,α(Ω) to the minimal solution of the boundary blowproblem
âu(x) = u(x)p(x) x â Ωu(x) = â x â âΩ.
iii) Assume that Îb = âΩc â© âΩ 6= â . Then, limun = â uniformly on compacts of Ωc âȘÎb.In addition, un converges in C2,α(Ω \ Ωc) to a solution of âu = up(x).
Theorem 2 says that problem (1.7) with f(x, u) = up(x) does not admit a solution if thecritical zone Ωc touches some component of âΩ. In this case un develops an âinfinite coreâin Ωc as n grows. On the other hand, un keeps finite in Ω+ = Ω \ Ωc and one wonderswhich boundary condition is satisfied by its limit u. Note that u solves the equation inΩ+. This discussion leads to a natural important problem: how does un behaves on theinterphase Îi when n ââ?
Our first result gives an answer to this question.
Theorem 3. Under the hypotheses of Theorem 2-iii) the solution un to (1.8) satisfies
lim un = âuniformly on Îi and hence such limit is also uniform in Ωc. In particular, u = limn un
in Ω+ solves the boundary blow-up problem âu = up(x) in Ω+, u = â on Îi âȘ Î+,Î+ = âΩ+ â© âΩ.
Remark 2. Similar conclusions on large solutions for a larger class of âvariable exponentâreactions are analyzed at the end of Section 5 (see Theorem 10).
Let us introduce next the corresponding features for the exponential nonlinearity f(x, u)= ep(x)u. First we state some known facts extracted from [10].
Theorem 4 ([10]). Assume that the domains Ωc â Ω are bounded, of class C2,α and thatΩc satisfies (H). Let p â Cα(Ω) be a nonnegative function, p 6⥠0, such that
Ωc = x â Ω : p(x) = 0.Then, problem
(1.9)
âu(x) = ep(x)u(x) x â Ωu(x) = n x â âΩ.
admits for each n â N a classical solution un â C2,α(Ω). Moreover,i) If Îb = â and thus Ωc â Ω then un converges in C2,α(Ω) to the minimal solution toâu = ep(x)u in Ω, u = â on âΩ.
ii) If on the contrary Îb 6= â then un â â uniformly in compacts of Ωc âȘ Îb while un
converges to a finite solution of the equation in Ω+ = Ω \ Ωc.
The behavior of the solution un to (1.9) at Îi was furthermore studied in [10] under therestrictive assumption that p(x) vanishes at Îi according to
(1.10) p(x) = o(d(x)),
as d(x) â 0 where d(x) = dist(x, Îi). Our second result just removes this restriction(compare with Lemma 3.4 in [3]).
MAXIMUM PRINCIPLE 5
Theorem 5. Assume the hypotheses of Theorem 4-ii) and let un be the solution to (1.9).Then,
lim un = âuniformly on the interphase Îi. Moreover, un â u in C2,α(Ω+) to the minimal solutionto âu = ep(x)u with boundary condition u = â on âΩ+ = ÎiâȘÎ+, where Î+ = âΩ+â©âΩ.
Now we change the scenario and turn our attention to population dynamics. Specifically,we deal with the problem,
(1.11)
ââu(x) = λm(x)u(x)â a(x)up(x) x â Ω
u(x) = 0 x â âΩ,
where p > 1 is constant, λ is a parameter and coefficients m, a â Cα(Ω), a(x) â„ 0, a 6= 0.When m = 1 and a(x) > 0 in Ω (or plainly, a = 1) (1.11) constitutes the classical logisticmodel where u is the equilibrium population density of the species, λ its the natural growthrate and a stands for the medium carrying capacity.
In problem (1.11) we are assuming that the critical domain Ωc â Ω is defined as
(1.12) Ωc = x â Ω : a(x) = 0.Thus a large amount of resources is available for the species u in Ωc. On the other hand, thecoefficient m(x) in (1.11) may change sign. Under these assumptions, a detailed analysisof problem (1.11) was performed in [5]. We use the notation λQ
1 (ââ + q) to designate theprincipal Dirichlet eigenvalue of (ââ+ q)u = ââu+ qu in a domain Q â RN (Section 2).
Theorem 6 ([5]). Under the previous assumptions on coefficients a,m suppose Ωc â Ωare C2,α bounded domains, Ωc being defined by (1.12).
Then (1.11) admits a positive solution if and only if
(1.13) λΩ1 (âââ λm) < 0 < λΩc
1 (âââ λm).
In this case, the positive solution is unique. Moreover, let (a, b) â R be a component of theset defined through (1.13) and let uλ be the solution associated to every λ â (a, b). Then
i) uλ bifurcates from zero at λ = λ0 â a, b provided λΩ1 (âââ λm)|λ=λ0
= 0, i.e.
limλâλ0
uλ = 0 in C2,α(Ω).
ii) uλ bifurcates from infinity at λ = λâ â a, b if λΩc1 (ââ â λm)|λ=λâ = 0 in the sense
that
(1.14) limλâλâ
|uλ|C(Ω) = â.
The singular behavior of uλ in (1.14) posed the problem of giving an explanation onhow singularities are developed by uλ so that it ceases to exist when λ crosses λâ. Toanswer this question was the main objective of [11], [18] which only dealt with the logisticproblem corresponding to (1.11) with m = 1. The existence range (1.13) reads in this case
λΩ1 := λΩ
1 (ââ) < λ < λΩc1 := λΩc
1 (ââ).
Bifurcation from infinity occurs at λ = λΩc1 . What is more important:
limλâλΩc
1
uλ = â,
uniformly in Ωc provided that the extra condition
(1.15) a(x) = o(d(x)) as d(x) â 0,
6 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
holds, where d(x) = dist(x,Îi) . Furthermore
limλâλΩc
1
uλ = u in C2,α(Ω+)
where u defines either the minimal solution to the blow-up problem ââu = λuâ a(x)up
in Ω+, u = â on âΩ+ if Ω+ â Ω or either the minimal solution to ââu = λuâ a(x)up inΩ+ and boundary conditions u = â on Îi, u = 0 on Î+ = âΩ+ â© âΩ.
It should be mentioned that some of these features of uλ were discovered also in [3]where, however, authors succeeded in suppressing restriction (1.15).
In our next result we extend the preceding analysis to problem (1.11) in the case wherethe coefficient m exhibits both signs. Notice that a consequence of losing the conditionm > 0 is that uλ ceases to be monotone in λ which is an important information in theanalysis in [11] and [18]. On the other hand, and as in [3], we are also removing therestriction (1.15).
Theorem 7. Suppose a,m â Cα(Ω), a â„ 0, a 6= 0, m 6= 0, while the subdomain Ωc â Ωis defined through (1.12) and satisfies (H). For every λ satisfying (1.13) let uλ â C2,α(Ω)designate the positive solution to (1.11). Then,
a) If m = 0 in Ωc the set (1.13) of âexistenceâ values of λ is either a single interval(λâ2, λâ1), λâ1 < 0 (alternatively, (λ1, λ2), λ1 > 0) or is the union of two intervals(λâ2, λâ1) âȘ (λ1, λ2), λâ1 < 0 < λ1. In all of these cases λ±2 = ±â.
b) If m 6= 0 in Ωc, inequalities (1.13) define either a finite interval (λ1, λ2) â R+
(alternatively, (λâ2, λâ1) â Râ) or is the union of two intervals (λâ2, λâ1) âȘ(λ1, λ2), λâ1 < 0 < λ1. In the latter case one (but not both at the same time !) ofthe λ±2 could be infinite.
c) uλ bifurcates from zero at the values λ = λ±1. Furthermore, if λ2 = â (respec-tively, λâ2 = ââ) then limλââ |uλ|C(Ω) = â (limλâââ |uλ|C(Ω) = â).
d) Assume λ2 is finite. Then
limλâλ2
uλ = â
uniformly in ΩcâȘÎi. Furthermore, there exists a subsequence λnâČ to every sequenceλn â λ2 such that lim uλnâČ = uâ in C2,α(Ω+), Ω+ = Ω \ Ωc where u = uâ definesa solution either to the problem
(1.16)
ââu(x) = λ2m(x)u(x)â a(x)up(x) x â Ω+
u(x) = â x â âΩ+,
if Ω+ â Ω, or to the problem
(1.17)
ââu(x) = λ2m(x)u(x)â a(x)up(x) x â Ω+
u(x) = â x â âÎi
u(x) = 0 x â âÎ+,
provided Î+ = âΩ+ â© âΩ is nonempty. Moreover, if a(x) satisfies the decay con-dition
(1.18) a(x) ⌠C0d(x)Îł ,
as d(x) = dist(x,Îi) â 0, for certain positive constants C0, Îł, then the strongerconvergence limλâλ2
uλ = u holds in C2,α(Ω+), where u is the unique solution toeither (1.16) or (1.17).
MAXIMUM PRINCIPLE 7
Of course, the same conclusions hold true at λ = λâ2 provided this value isfinite.
Remark 3. It should be remarked that λ = λ±1 provide with the so-called principaleigenvalues of the weighted eigenvalue problem
ââu(x) = λm(x)u(x) x â Ωu(x) = 0 x â âΩ.
At least one of these numbers always exists (see [16] for a detailed analysis on this issue).
Finally, we study a nonlinear problem that is an extension of
(1.19)
ââu(x) = λu(x)â a(x)up(x) x â Ω
u(x) = 0 x â âΩ,
where λ is a parameter, a, p â Cα(Ω) with a positive in Ω while the variable reaction orderp satisfies p(x) â„ 1, p(x) 6⥠1 in Ω. Solutions to (1.19) constitute equilibrium states of areaction-diffusion process governed by a reaction characterized by a variable stoichiometricexponent p(x).
In the framework of this problem the critical domain Ωc is defined as
(1.20) Ωc = x â Ω : p(x) = 1,where it is understood that Ωc is a nonempty C2,α bounded subdomain of Ω.
Under the precedent conditions, problem (1.19) was proposed and studied in detail in[17]. We are now simplifying the proof of the basic facts of that work and at the sametime extending its scope in several issues. First, by adding âin the line of problem (1.11)âa sign-indefinite weight m(x) in front of λ (which causes the loss of monotonicity of uλ
in λ). Second, by allowing more general configurations in Ωc (in [17], Ω+ = Ω \ Ωc isconstrained to satisfy Ω+ â Ω). Last, but not least, we provide the proof of a missingstep in [17]. Theorem 1 is just in the kernel of this point and has to do with the behaviorof solutions on the interphase Îi (Remark 5-b)).
Our results are collected in the following statement.
Theorem 8. Consider the problem
(1.21)
ââu(x) = λm(x)u(x)â a(x)u(x)p(x) x â Ω
u(x) = 0 x â âΩ,
where coefficients a(x) > 0 and p(x) â„ 1 satisfy the preceding hypotheses, Ωc is defined in(1.20) and satisfies (H), while m â Cα(Ω) is arbitrary regarding sign. Then, the followingproperties hold.
i) Problem (1.21) admits at most one positive (classical) solution. Such a solution exists ifand only if
(1.22) λΩ1 (âââ λm + aÏΩc) < 0 < λΩc
1 (âââ λm + a),
where ÏΩc is the characteristic function of Ωc. The solution to (1.21) is denoted by uλ.ii) If m = 0 in Ωc, the existence region (1.22) for λ is a single interval (λâ2(m), λâ1(m))(respectively, (λ1(m), λ2(m))) or is either a union (λâ2(m), λâ1(m)) âȘ (λ1(m), λ2(m))where in all cases λâ1(m) < 0 < λ1(m) and λ±2 = ±â. If otherwise, m 6= 0 in Ωc thenthe set (1.22) is either a finite interval (λ1(m), λ2(m)) (respectively, (λâ2(m), λâ1(m)))or splits as (λâ2(m), λâ1(m)) âȘ (λ1(m), λ2(m)). In all cases λâ1(m) < 0 < λ1(m) whilein the latter option one (but not both) of the values λ±2 could be infinite.
8 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
iii) uλ bifurcates from zero at λ = λ1(m) (respectively, λ = λâ1(m)). In addition |uλ|C(Ω) ââ as λ â λ2(m) (respectively, λ â λâ2(m)) provided λ2(m) = â (λâ2(m) = ââ).
iv) Assume λ2(m) is finite. Then
limλâλ2(m)
uλ = â
uniformly on compacts of Ωc âȘ Îi. Furthermore, for any sequence λn â λ2 there exists asubsequence λnâČ and a solution uâ â C2,α(Ω+) to the blow-up problem
(1.23)
ââu(x) = λ2(m)m(x)u(x)â a(x)u(x)p(x) x â Ω+
u(x) = â x â âΩ+,
if Ω+ â Ω, or either to the mixed blow-up problem
(1.24)
ââu(x) = λ2(m)m(x)u(x)â a(x)u(x)p(x) x â Ω+
u(x) = â x â Îi
u(x) = 0 x â Î+,
if Î+ = âΩ+ â© âΩ is nonempty, such that
uλnâČ â uâ,
in C2,α(Ω+).v) If λ2(m) is finite and in addition m â„ 0 in Ω+ then uλ â u in C2,α(Ω+) where u is theminimal solution to (1.23) if Ω+ â Ω, or either the minimal solution to (1.24) providedÎ+ 6= â .
Analogous features as in iv)-v) hold true as λ â λâ2(m) provided λ = λâ2(m) is finite.
This work is organized as follows: Section 2 contains the material on the maximumprinciple; boundary blow-up problems are considered in Sections 3; population dynamics(problem (1.11)) is studied in Section 4 and finally, Section 5 is devoted to the variableexponent problem (1.21).
2. On the maximum principle
As we have mentioned in the introduction, a well-known and weaker version of themaximum principle is as follows: if u â C2(Ω) â©C(Ω) satisfies (L+ c)u â„ 0 in Ω (L as inSection 1), with c â Lâ and nonnegative in Ω, then
(2.1) infΩ
u â„ infâΩ
uâ
(see [12]). On the other hand, the most immediate example, say (L+ c)u = âuâČâČ + 4u inΩ = (âÏ/2, Ï/2) with u(x) = sin(2x) + 2 satisfying (L+ c)u â„ 0, shows that (1.2)-(1.3) isfalse for arbitrary zero order perturbations of the operator L.
Nevertheless, such simple statement is enough to ensure the existence of a principal(Dirichlet) eigenvalue of L+ c. That means a real value of λ such that the problem
(L+ c)u(x) = λu(x) x â Ωu(x) = 0 x â âΩ
admits a positive solution. More precisely, a unique such principal eigenvalue exits âregardless the sign of c â provided that Ω is of class C2,α and aij , bi, c â Cα(Ω) (see[14], [16], and weaker versions in [2], [4], [6]). Furthermore, such eigenvalue âwhich will
MAXIMUM PRINCIPLE 9
be denoted as λΩ1 (L + c)â is simple, increasing in c and decreasing with respect Ω, i.e.
Ω0 â Ω, Ω0 6= Ω impliesλΩ
1 (L+ c) < λΩ01 (L+ c).
As additional relevant features to be used in this work, it also holds that λΩ1 (L + c) is
concave in c ([2], [14], [16]), the function E(t) := λΩ1 (L+ tc) is real analytic for t â R and
λΩ1 (L+ c) is continuous and smooth with respect to smooth perturbations of the domain
Ω ([14], [16] and [13]).A still weaker result than the previous ones which is often termed as the maximum
principle is the following: a function u â C2(Ω) â© C(Ω) turns out to be nonnegativein Ω when (L + c)u â„ 0 in Ω together with u â„ 0 on the boundary âΩ. For instancec â„ 0 implies, via (2.1), the maximum principle in this sense. On the other hand, suchstatement, which largely suffices for most of applications âe. g. the comparison principleâis elegantly characterized by the principal eigenvalue as the following result asserts ([2],[14], [16], [4] and also [6] for its extension to the p-Laplacian operator).
Theorem 9. Suppose that Ω is a class C2,α bounded domain, L + c satisfies (1.1) andhas coefficients in Cα(Ω). Then, the following statements are equivalent.
i) λΩ1 (L+ c) > 0.
ii) Every u â C2(Ω) â© C(Ω) such that (L + c)u â„ 0 in Ω and u â„ 0 on âΩ satisfiesu â„ 0 in Ω.
However, an elementary example shows that (2.1) could be false if it is only assumedthat λΩ
1 (L+ c) > 0. In fact, consider (L+ c)u = âuâČâČâ Δu, 0 < Δ < 1 in Ω = (âÏ/2, Ï/2).If u(x) = x2 â k and k â„ Ï2
4 + 2Δ then (L+ c)u ℠0 but (2.1) fails.
The spirit of Theorem 1 is just to partially recover (1.2)-(1.3) under the assumptions ofTheorem 9. As will be seen later, this result will be instrumental in our applications.
Proof of Theorem 1. The positivity of λ1 implies that the problem
(L+ c)u(x) = f(x) x â Ωu|âΩ = 0,
admits, for each f â Cα(Ω), a unique solution in C2,α(Ω) ([12]). This shows the existenceof Ï which is, by Theorem 9, nonnegative. By choosing M large we find (L+(c+M))Ï â„ 0with c + M â„ 0 and the strong maximum principle ([12]) yields Ï > 0 in Ω.
To show ii) we use ideas from [20] (see Theorem 10, Chap. 2). Introduce the newunknown v as u = Ïv. Then,
L1v â„ 0
in Ω with L1v = ââij aijâijv +
âibiâ 2
Ï
âj aijâjÏâiv. Then, the maximum principle
(1.2)-(1.3) impliesinfΩ
v = infâΩ
v,
which is (1.4). €
Remarks 4.a) A direct application of the strong maximum principle shows that
u(x) = infâΩ
uÏ(x)
holds for all x â Ω if such equality is attained somewhere in Ω.
10 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
b) Assume that c â„ 0, c 6= 0 in Theorem 1. Then 0 < Ï < 1 in Ω and âÏâÎœ > 0 in âΩ,
being Îœ the outward unit normal at âΩ. In this case, setting u = infâΩ u with u 6= uÏ andu(x) = u for x â âΩ one finds
âu
âÎœ< u
âÏ
âÎœ,
at x = x. For u †0 this is, of course, Hopfâs Lemma ([12]).c) In the framework of strong solutions a lower degree of smoothness in the coefficients ofL+ c is required for Theorem 1 to hold. In fact, suppose that (1.1) holds with aij â C(Ω),bi â Lâ(Ω), c â Lâ(Ω) while Ω â RN is bounded and C2. Then, the operator L+ c hasa principal eigenvalue λΩ
1 (L + c) with a positive associated eigenfunction Ï â W 2,q(Ω) â©C1,ÎČ(Ω) for all q > 1, 0 < ÎČ < 1. In addition, λΩ
1 (L+ c) exhibits the same features as inthe classical setting (see [2]).
The positivity of λΩ1 (L + c) characterizes the maximum principle for strong solutions.
Namely, (L+ c)u â„ 0 in Ω for u â W 2,qloc (Ω) â©C(Ω), q â„ N , and u â„ 0 on âΩ implies that
u ℠0 in Ω (see [2] where the regularity of Ω is considerably relaxed).Under the preceding conditions and assuming λΩ
1 (L + c) > 0, Theorem 1 now statesthat if u â W 2,q
loc (Ω) ⩠C(Ω), q ℠N , satisfies
(L+ c)u ℠0 in Ω,
thenu(x) â„ inf
âΩuÏ(x) x â Ω,
where Ï is the solution to (L+ c) Ï(x) = 0 x â Ω
Ï(x) = 1 x â âΩ.
Note that Ï is positive in Ω with Ï â W 2,q(Ω) â© C1,ÎČ(Ω) for all q > 1, 0 < ÎČ < 1.
3. Boundary blow-up problems
In this section we give the proofs of Theorems 3 and 5 concerning boundary blow-upproblems and large solutions.
Proof of Theorem 3. Let un â C2,α(Ω) be the solution to
âu(x) = u(x)p(x) x â Ωu(x) = n x â âΩ.
It follows from Theorem 2 that un ââ uniformly in compacts of ΩcâȘÎb. In fact, a directproof of this fact runs as follows: un â„ nwn where w = wn solves the problem
âw(x) = np(x)â1(w(x))p(x) x â Ωc
w(x) = 1 x â Îb
w(x) = 0 x â Îi.
The assertion is then a consequence of the fact that wn â w0 in C1,ÎČ(Ω) where w0 is thepositive strong solution to
âw(x) = Ïp=1(w(x))p(x) x â Ωc
w(x) = 1 x â Îb
w(x) = 0 x â Îi.
MAXIMUM PRINCIPLE 11
To describe the behavior of un in Ω+ = Ω \ Ωc choose a component Ω+0 of Ω+. Then
un †u in Ω+0 where u = u â C2,α(Ω+
0 ) is the minimal solution to
âu(x) = u(x)p(x) x â Ω+0
u(x) = â x â âΩ+0 ,
the existence of such solution is proved in [9]. Now, the increasing character of un togetherwith that estimate is enough to ensure that un converges in C2(Ω) towards a solution u ofthe equation. Regarding the boundary conditions, it is evident that u = â on âΩ+
0 â© âΩif such part of âΩ+
0 is nonempty. Our main objective now is to show that
limun = â,
uniformly on each component Î0,i of âΩ0 ⩠Ω. Being Îi the disjoint union of such com-ponents, this proves Theorem 3. To fulfill this objective we use the approach in [3]. Letxn â Î0,i be such that
un := un(xn) = infÎ0,i
un.
Let us prove that un â â. Otherwise, un = O(1). Assume this and proceed with thefollowing constructions. First, consider the strip
U+ = x â Ω+ : dist(x, Î0,i) < ηwhere η > 0 is small enough. In U+ consider the problems
âu(x) = u(x)p(x) x â U+
u(x) = un(x) x â âU+,
âu(x) = u(x)p(x) x â U+
u(x) = un(x) x â âU+ ⩠Ω+
u(x) = un x â Î0,i.
Both of them have a unique positive solution, un is just the solution of the first one whilewe set u := un the solution of the latter one. We have that un â„ un and therefore,
(3.1)âun
âÎœ(xn) †âun
âÎœ(xn),
for all n, where Îœ is the outward unit normal to U+ on Î0,i. Since both un and un(·)|âU+â©Î©+
are bounded, it follows that un = O(1) in C2,α(U+). Thus, from (3.1) we get that âunâÎœ (xn)
is bounded above
(3.2)âun
âÎœ(xn) †C.
Now we proceed to estimate âunâÎœ (xn) from below. To this end set
Uâ = x â Ωc : dist(x,Î0,i) < η,with η > 0 so small as to have U
â \ Î0,i â Ωc. Pick M â„ 1 such that Mu1 â„ 1 in Uâ.Then w = Mun solves
âw(x) = a(x)(w(x))p(x) x â Uâ
w(x) = Mun(x) x â âUâ,
with a(x) = Mp(x)â1. Hence, w = Mun satisfiesââw(x) + âaâw(x) â„ 0 x â Uâ
w(x) = Mun(x) x â âUâ,
12 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
with âaâ = supUâ a. Since λUâ1 (ââ + âaâ) > 0, then Theorem 1 applied in Uâ yields
(3.3) infUâ
vn = infâUâ
vn
where vn = Mun/Ï (see Theorem 1 for the definition of Ï). Since vn ââ uniformly onâUâ ⩠Ωc we get
infUâ
vn = infÎ0,i
vn = vn(xn) = Mun.
We now prove that
(3.4) limâvn
âÎœ |x=xn
= â,
where Îœ = Îœ(x) is the inner unit normal to Uâ at x â Î0,i. Once (3.4) is proved weconclude
lim Mâun
âÎœ(xn) = limun
âÏ
âÎœ(xn) + lim
âvn
âÎœ(xn) = â,
since un = O(1). This contradicts the previous estimate of such derivative, (3.2). There-fore
lim un = âuniformly on Î0,i and, for the same reasons, on the whole of Îi.
Now to show (3.4) consider the family of annuli
An = x :η
8< |xâ yn| < η
4, yn = xn +
η
4Μ(xn).
After decreasing η if necessary, it holds that An â Uâ while âAn â© Î0,i = xn. Forimmediate use set â1An = x : |xâ yn| = η
8, â2An = x : |xâ yn| = η4 the components
of âAn. Observe that v = vn satisfiesLv(x) â„ 0 x â An
v(x) = vn(x) x â âAn,
with Lv = ââv â 2Ïâ1âÏâv. Moreover,
infAn
vn = vn(xn) = Mun,
since infâ1An vn ââ.As in [12], we introduce now the usual barrier h(r) = eΞ(( η
4)2âr2)â1, r = |xâyn|, where
Ξ > 0 can be chosen independent of n so that
Lh < 0 x â An,
for every n. Thus,L(vn âMun â knh) â„ 0 x â An,
for kn â„ 0. By setting
kn =infâ1An vn âMun
h(η/8)we finally obtain,
vn â„ Mun + knh x â An,
since such inequality holds on âAn. Therefore,
limâvn
âÎœ(xn) â„ lim kn
(âhâČ
(η
4
))= â.
MAXIMUM PRINCIPLE 13
Finally, once it has been shown that un ââ uniformly on Îi this uniformity propagatesto Ωc. In fact, one obtains from (3.3) that such convergence is uniform in a closed stripx â Ωc : dist(x,Îi) †η. €
Now we prove Theorem 5.
Proof of Theorem 5. Let Î0,i any component of Îi which, for simplicity, will be denotedas Îi. To prove the uniform limit limun = â on Îi we set,
infÎi
un = un(xn) := un xn â Îi
and assume, arguing by contradiction, that un = O(1). We then proceed with a similarconstruction as in Theorem 3. We introduce U+ = x â Ω+ : dist(x,Îi) < η, η > 0small, where we consider the problems
âu(x) = ep(x)u(x) x â U+
u(x) = un(x) x â âU+,
âu(x) = ep(x)u(x) x â U+
u(x) = un(x) x â âU+ ⩠Ω+
u(x) = un x â Îi.
Thus we getun(x) â„ un(x) x â U+,
where u = un stands for the solution to the latter problem. That is whyâun
âΜ†âun
âÎœ
at x = xn, with Îœ the outward unit normal. Since un stays bounded on âU+ ⩠Ω+
the C2,α(U+) norm of un is bounded and this means that the normal derivative âun/âÎœevaluated at x = xn remains bounded from above.
On the other hand, u = un solvesâu(x) = 1 x â Uâ
u(x) = un(x) x â âUâ,
with Uâ = x â Ωc : dist(x,Îi) < η and η > 0 small enough. If z = Ï(x) stands for thesolution of ââz = 1 in Uâ with z|âUâ = 0, then v = vn := un + Ï is the solution to
âv(x) = 0 x â Uâ
v(x) = un(x) x â âUâ.
ThusinfUâ
vn = vn(xn) = un
for large n, since vn â â uniformly on âUâ ⩠Ωc. By introducing now the family An ofannuli constructed in the proof of Theorem 3 and employing the same barrier h(r) we get
un â„ un + knhâ Ï x â An,
with kn â â. This in turns implies that âunâÎœ |x=xn
â â which contradicts the previousestimate. Therefore un ââ uniformly on Îi. Finally,
infUâun + Ï â„ inf
Îi
un
implies that un ââ in the whole of Uâ which, combined with Theorem 4, yields un ââ
uniformly in Ωc. €
14 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
4. A Population dynamics problem with a two-signed weight
This section is devoted to prove Theorem 7.
Proof of Theorem 7. Let us begin with a)-b) and recall that m 6= 0. We are only consid-ering b) since the analysis of a) is much simpler.
If m â„ 0, m 6= 0 in Ωc then λ0(t) = λΩc1 (ââ â tm) is real-analytic, non increasing
and concave in R (Section 2). In addition λ0(0) > 0. Therefore, λ0(t) is decreasing withlimtââ λ0(t) = ââ since the derivative of λ0(t) must be negative somewhere. Thusλ0(t) > 0 = (ââ, t2) with t2 > 0. Symmetrically, λ0(t) > 0 = (tâ2,â) with tâ2 < 0if m †0, m 6= 0 in Ωc, while λ0(t) must vanish at t = t±2.
Suppose now that m takes the two signs in Ωc. By taking a ball B+ â B+ â m >
0 ⩠Ωc and observing that
λΩc1 (âââ tm) < λB+
1 (âââ tm) t â R,
it follows that limtââ λ0(t) = ââ. An analogous argument using a ball Bâ â Bâ â
m < 0 ⩠Ωc gives limtâââ λ0(t) = ââ. Thus
λ0(t) > 0 = (tâ2, t2) tâ2 < 0 < t2,
since λ0(0) > 0. In addition, λ0(t) vanishes at t = t±2.By setting λ(t) = λΩ
1 (ââ â tm), the previous arguments prove that λ(t) < 0 =(ââ, t1) with t1 > 0 if m â„ 0 (respectively, λ(t) < 0 = (tâ1,â, ), tâ1 < 0 when m †0in Ω) or, provided m exhibits both signs in Ω,
λ(t) < 0 = (ââ, tâ1) âȘ (t1,â),
with tâ1 < 0 < t1. In all cases λ(t) vanishes at t = t±1. Finally, the assertions in b) followfrom the preceding discussion and the fact that λ(t) < λ0(t) in R.
Regarding c), and as was quoted in Theorem 6-i), we refer to [5] for the proof that uλ
bifurcates from zero at λ = λ±1 (see also Section 5). On the other hand, suppose thatλ2 = â. According to the preceding discussion this means that m †0 in Ωc while m
attains a positive value somewhere in Ω+. Thus, a ball B+ â B+ â Ω+ can be found so
that m â„ m0 > 0 in B+. Let us denote as v = vλ the unique positive solution toââv(x) = λm(x)v(x)â a(x)(v(x))p x â B+
v|âB+ = 0.
Then we have, on one hand, that uλ ℠vλ in B+ for large λ. On the other hand, it iswell-known that (see the proof of Theorem 8)
limλââ
λâ 1
pâ1 vλ(x) > 0 x â B+.
Thus, it follows that |uλ|C(Ω) ââ as λ ââ.Let us assume next that, say λ2 < â. Then, it was shown in [5] that |uλ|C(Ω) ââ as
λ â λ2. We are going now quite beyond this result and show that
(4.1) uλ ââ as λ â λ2,
uniformly on compacts of Ωc. We first prove the existence of a function v â C1(Ω),v(x) > 0 in Ω, âv
âÎœ < 0 on âΩ, Îœ being the outer unit normal, such that
(4.2) uλ â„ v x â Ω,
MAXIMUM PRINCIPLE 15
for all λ2 â ÎŽ †λ †λ2 with ÎŽ > 0 small. In fact, by continuity a certain positive ”0
exists such that λΩ1 (ââ â λm) †â”0 in [λ2 â ÎŽ, λ2]. Setting Ï1 = Ï1(·, λ) the principal
(positive) eigenfunction associated to λΩ1 (âââλm) normalized so that |Ï1|C(Ω) = 1, then
it be can checked that u = ΔÏ1 is a subsolution to (1.11) with λ â [λ2 â ÎŽ, λ2] provided
Δpâ1 supΩ
a †”0.
Now, the continuity of Ï1 with respect to λ, when regarded in C1(Ω), and the fact thatâÏ1
âÎœ < 0 for every λ â [λ2 â ÎŽ, λ2] allow us to find v â C1(Ω), 0 < v < Ï1 in Ω for allλ â [λ2 â ÎŽ, λ2], with âv
âÎœ < 0. To complete the proof of the assertion observe that (1.11)admits arbitrarily large supersolutions in the range λΩc
1 (âââ λm) > 0 (see [5] or Section5). Therefore, inequality (4.2) follows.
To show (4.1) we deal with the case Îb 6= â since the case Ωc â Ω is much simpler. Tobegin with, take λn â λ2, λ2â ÎŽ †λn †λ2, put un = uλn and observe that u = un solves
ââu(x)â λnm(x)u(x) = 0 x â Ωc
u(x) = 0 x â Îb
u(x) = un(x) x â Îi.
We claim the existence of ηn > 0, ηn â 0, such that λUηn1 (âââλnm) = 0 where, for t > 0
we set Ut = x â Ω : dist(x,Ωc) < t. Then, taking Ï the positive eigenfunction associatedto λΩc
1 (ââ â λ2m), |Ï|C(Ω) = 1, and designating by Ïn the corresponding eigenfunction
associated to λUηn1 (ââ â λnm) it follows that Ïn â Ï in C2,α(Ωc) (see, for instance [13]
for a proof of this convergence).Let us choose K â Ωc âȘ Îb, compact, and any M > 0. We will follow the approach in
[3] to show that
(4.3) un(x) ℠ΞMÏ(x) x â K,
Ξ = 12 infK Ï, for large n. This certainly proves (4.1). In fact, set Ωc,η = x â Ωc :
dist(x,Îi) > η and select η > 0 small such that K â Ωc,η. If we take Îi,η = x â Ωc :dist(x,Îi) = η = âΩc,η ⩠Ωc, we can find η small such that
M supÎi,η
Ï â€ 12
infÎi,η
v.
Thus we find thatM sup
Îi,η
Ïn †infÎi,η
v.
for n â„ nM . Moreover
(4.4) MÏn(x) â„ 12
infK
Ï x â K,
for n â„ nK,M . We next observe that u = un satisfies
(4.5)
ââu(x)â λnm(x)u(x) = 0 x â Ωc,η
u(x) = 0 x â Îb
u(x) = un(x) x â Îi,η,
meanwhile un = MÏn is clearly a subsolution to (4.5). Since λΩc,η
1 (âââλnm) > 0 for alln, direct comparison gives
un(x) â„ MÏn(x) x â Ωc,η,
16 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
which, combined with (4.4) provides (4.3).As for the claim, take η0 > 0 small such that λ
Uη01 (âââ λ2m) < 0. The continuity of
the principal eigenvalue on the zero order term (Section 2) shows that λUη01 (âââλm) < 0
for all λ2 â ÎŽ < λ < λ2, ÎŽ > 0 small. On the other hand, λUt1 (ââ â λm) is smooth on
t â [0, η0] (see [13] and close results in [18]). That is why, for fixed λn close to λ2, a valueηn with 0 < ηn < η0 exists such that λ
Uηn1 (âââ λnm) = 0, what proves the claim.
We now show that limλâλ2uλ = â uniformly on Îi. To this end let us prove this limit
uniformly on each component of Îi. For simplicity we denote again Îi such component,we take λn â λ2, set un = uλn and suppose that
un = infÎi
un = un(xn) ââ xn â Îi,
to get a contradiction. Using the same arguments as in Section 2 we observe that
vn(x) †un(x) x â U+, U+ = x â Ω+ : dist(x,Îi) < η,η > 0 small, where v = vn solves
(4.6)
ââv(x) = λnm(x)v(x)â a(x)(v(x))p x â U+
v(x) = un x â Îi
v(x) = un(x) x â âU+ ⩠Ω+.
Then we getâun
âΜ†âvn
âÎœat x = xn,
with Îœ the outer unit normal to U+. Furthermore it holds that,âun
âÎœ(xn)
is bounded above. This follows from the fact that |vn|C2,α(U+
)is bounded. In fact such
assertion is a consequence of Schauderâs estimates applied to (4.6), after obtaining boundsof |un|C2,α(U
+)
near âU+ ⩠Ω+, and further getting bounds of |vn|C(U+
)and |vn|Cα(U
+).
To estimate un we notice that un †wn with w = wn the solution toââw(x) = λ2m
+(x)w(x)â a(x)(w(x))p x â Ω+
w(x) = un(x) x â âΩ+.
Thus un †uâ in U+ where uâ â C2,α(Ω+) is the minimal solution to
(4.7)
ââu(x) = λ2m
+(x)u(x)â a(x)(u(x))p x â Ω+
u(x) = â x â âΩ+.
This gives us Lâloc estimates of un which in turns lead to estimates in C2,α(Q) for everysubdomain Q â Q â Ω+ (see for instance [8]). On the other hand, the estimate vn †vwith v the solution of
ââv(x) = λ2m+(x)v(x)â a(x)(v(x))p x â U+
v(x) = supun x â Îi
v(x) = uâ(x) x â âU+ ⩠Ω+,
proves the boundedness of |vn|C(U+
). Going back to (4.6) we can now use the standard
W 2,q estimates to get that |vn|C1,α(U+
)†C, C a constant, for all 0 < ÎČ < 1, and that is
just what we wanted to obtain.
MAXIMUM PRINCIPLE 17
Our objective now is to show the divergence of the normal derivative âunâÎœ |x=xn
. Ac-cordingly, we take Uâ = x â Ωc : dist(x,Îi) < η, η > 0 small enough, and observe thatu = un solves,
ââu(x)â λnm(x)u(x) = 0 x â Uâ
u(x) = un(x) x â âUâ.
In order to apply Theorem 1 notice that λUâ1 (ââ â λnm) > 0 and define Ï = Ïn the
solution to ââÏ(x)â λnm(x)Ï(x) = 0 x â Uâ
Ï(x) = 1 x â âUâ.
The fact that λUâ1 (âââ λ2m) > 0 and the continuity of λUâ
1 (âââ λm) > 0 on λ allowto conclude the uniform boundedness of Ïn in the norm of C2,α(Uâ). On the other handTheorem 1 ensures that
wn =un
Ïn
satisfies
(4.8) infUâ
wn = wn(xn) = un,
for large n due to the divergence of wn when regarded on âUâ â©Î©c. We have in additionthat w = wn solves
Lnw(x) = 0 x â Uâ
w(x) = un(x) x â âUâ,
where the operator Lw = ââw â 2âÏn
Ïnâw has its coefficients uniformly bounded in
Cα(Uâ). The same argument as in the proof of Theorem 3 allows us to write
wn(x) â„ un + knh(|xâ yn|) η
8< |xâ yn| < η
4,
with yn = xn +(η/4)Îœ(xn), Îœ the inner unit normal to Uâ, h = h(r) is the uniform barrierintroduced in Theorem 3 and kn is a positive sequence that can be chosen so that kn ââ.
The inequality above leads to the fact that âwnâÎœ (xn) ââ and finally
limâun
âÎœ(xn) â„ lim
âwn
âÎœ(xn) + limun
âÏn
âÎœ(xn) = â.
This is the contradiction that we look for and hence the proof that uλ ââ uniformly onÎi is concluded. Notice that this divergence propagates to the whole of U
â through (4.8).This means that uλ ââ uniformly in compacts of Ωc âȘ Îi.
To finish the proof of Theorem 7 let us analyze the behavior of uλ in Ω+. Assume forsimplicity that Ωc â Ω (and thus Î+ = âΩ+ â© âΩ = â ). Estimating un by the solutionu = uâ of (4.7) we get âas was already shownâ C2,α
loc estimates of un in Ω+. Thus, asubsequence can be extracted from un (we still use un to designate such subsequence)such that un â uâ in C2(U+) and so u = uâ solves
ââu(x) = λ2m(x)u(x)â a(x)(u(x))p
in Ω+. However we assert that for every M > 0 it holds that
(4.9) uâ(x) â„ uM (x) x â Ω+,
18 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
where u = uM is the solution toââu(x) = λ2m(x)u(x)â a(x)(u(x))p x â Ω+
u(x) = M x â âΩ+.
In fact, it can be shown that uM = limuM,n in C2,α(Ω+) where u = uM,n is, in turn, thesolution to
ââu(x) = λnm(x)u(x)â a(x)(u(x))p x â Ω+
u(x) = M x â âΩ+.
Since un â„ uM,n for large n we readily conclude (4.9). But this implies that
uâ â„ u
where u is the minimal solution to
(4.10)
ââu(x) = λ2m(x)u(x)â a(x)(u(x))p x â Ω+
u(x) = â x â âΩ+,
whose existence is well-known (see [8]). Thus uâ solves the blow-up problem (4.10). More-over, it is also known that under condition (1.18), problem (4.10) only admits a uniquesolution and so uâ = u (see [8]). This implies in particular that the whole sequenceun converges to u in C2,α(Ω+). Finally, to deal with the case Î+ 6= â we only need toadd the boundary condition u = 0 on Î+ in all of the involved auxiliary boundary valueproblems. €
5. A reaction-diffusion problem involving a variable reaction
Proof of Theorem 8. Let us begin with the existence and uniqueness questions raised in i).Uniqueness can be obtained by using, among several approaches, the sweeping principle(see [22]). Namely, let u1, u2 â C2,α(Ω) be positive solutions to (1.21) and set tâ â„ 1 theinfimum among the values t â„ 1 such that tu2 â„ u1. By the strong maximum principleand the fact that tu2 defines a supersolution when t â„ 1, it can be shown that tâ = 1.This shows that u2 â„ u1 and the complementary inequality is obtained in the same way.
To show the necessity of (1.22), the existence of a positive solution u â C2,α(Ω) impliesthat, â«
Ω|âu|2 â λmu2 + aÏΩcu
2 < 0.
The variational characterization of the first eigenvalue implies
λΩ1 (âââ λm + aÏΩc) < 0.
On the other hand, such solution u is a classical positive supersolution to ââuâ λmu +au = 0 in Ωc which is positive in Îi = âΩc ⩠Ω. As it is well-known ([5], [16], [6]) thisimplies the positivity of λΩc
1 (âââ λm + a).Concerning the existence of solutions under (1.22), we proceed in a direct way instead of
using the involved perturbation approach developed in [17] (which is in addition restrictedto the case m = 1 and Ω+ â Ω). The crucial point is to find a subsolution. Notice thatthe natural candidate, namely the eigenfunction Ï to
(âââ λm + aÏΩc)Ï = ÏÏ x â Ω, Ï|âΩ = 0,
Ï = λΩ1 (âââλm+aÏΩc), must be ruled out. Indeed the standard choice u = Î”Ï requires
âÏ â„ a(ΔÏ)p(x)â1
MAXIMUM PRINCIPLE 19
in Ω. However, and no matter how small Δ > 0 is taken, such inequality can not be givenfor granted at Îi since p = 1 there.
To avoid this problem and construct a subsolution we introduce Q = B(Ωc, ÎŽ) ⩠Ω =x â Ω : dist(x,Ωc) < ÎŽ with ÎŽ > 0 small and observe that with a suitable election of ÎŽwe have
Ï := λΩ1 (âââ λm + aÏQ) < 0.
Setting Ï â W 2,q(Ω) â© C1,ÎČ(Ω), q > 1, 0 < ÎČ < 1 arbitrary, the positive eigenfunctionassociated to Ï
(âââ λm + aÏQ)Ï = ÏÏ x â Ω, Ï|âΩ = 0,
|Ï|C1(Ω) = 1 (see Remark 4-c)), then u = Î”Ï with Δ > 0 small, gives a subsolution to(1.21). In fact, we need
(5.1) (âââ λm + aÏΩc)u †aÏΩ+up(x) x â Ω.
In Ωc this means(âââ λm + aÏΩc)u †0,
which holds due to the negativity of Ï. In Q \ Ωc (5.1) reads as
âÏ + aÏQ\Ωcâ„ aÏQ\Ωc
up(x)â1,
which is certainly true provided 0 < Δ †1. In Ω \Q (5.1) is equivalent to
(5.2) â Ï â„ aup(x)â1.
Such inequality is achieved for Δ small either if Î+ = âΩ+ â© âΩ is empty or provided thatp(x) > 1 for all x â Î+ if Î+ 6= â . However, in the latter case (5.2) could become falseif p(x) = 1, x â Î+ and x â x. To tackle the problem if this is the case we redefine thesubsolution. By introducing
ΩâČ = x â Ω : dist(x,Î+) > ÎŽ1,ÎŽ1 > 0 can be taken so small as to have
ÏâČ := λΩâČ1 (âââ λm + aÏQ) < 0.
If ÏâČ stands for the corresponding associated positive eigenfunction in ΩâČ, |ÏâČ|C(Ω
âČ)= 1, and
ÏâČ is its extension by zero to Ω, then u = ΔÏâČ â H1(Ω)â©Cα(Ω) defines a weak subsolution(which still enables us to get a classical solution to (1.21)). Indeed, we succeed now ingetting (5.2) by setting Δ > 0 small. This finishes the construction of the subsolution.
To find out a comparable supersolution we observe, following [5], that a positive largeenough function v â C1(Ω), v|âΩ = 0, can be found so that
λΩ1 (âââ λm + aÏΩc + aÏΩ+vp(x)â1) > 0.
Taking Ï â W 2,q(Ω)â©C1,ÎČ(Ω) its associated positive eigenfunction with |Ï|C(Ω) = 1, thenu = MÏ defines a supersolution provided M â„ 1 is taken so large as to have MÏ â„ v.
The proof of assertion ii) follows exactly with the same arguments used in the corre-sponding proof of a) and b) in Theorem 5. Thus, let us show assertion iii) and begin bythe bifurcation at, say λ = λ1. First, we have that there exists C > 0 such that
|uλ|C(Ω) †C,
for λ1 < λ †λ1+Ύ. This is a consequence of the existence of arbitrarily small subsolutionsand the existence of a uniform supersolution u1 in that interval. In fact, since
λΩc1 (âââ λ1m + a) > 0
20 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
the previous construction proves the existence of u1 such that u1|âΩ = 0 together with
ââu1 â λ1mu1 + au1p(x) ℠Ξu1 x â Ω,
for a certain Ξ > 0. Thus,
ââu1 â λmu1 + au1p(x) â„ 0 x â Ω,
if 0 < (λ â λ1) sup |m| †Ξ. Once we have Lâ bounds of uλ as λ â λ1 we are pro-gressively getting, as it is well-known, uniform bounds in W 2,q(Ω), C1,ÎČ(Ω) and finally inC2,α(Ω). Then, after passing through subsequences if necessary, and taking into accountthe nonexistence of positive solutions at λ = λ1 one concludes that uλ â 0 in C2,α(Ω) asλ â λ1.
Next, we describe the limit as, λ â â, that is, when λ2(m) = â (notice that thisrequires m †0 in Ωc together with the positiveness of m somewhere in Ω). Thus, thereexists a ball B+ â Ω+ where m is bounded away from zero. Consider the problem
(5.3)
ââv(x) = λm(x)v(x)â a(x)(v(x))p(x) x â B+
v|âB+ = 0.
Our previous study shows that (5.3) admits a unique positive solution v = vλ for allλ > λ1,B+ , t = λ1,B+ being the unique root of λB+
1 (ââ â tm) = 0. Moreover, vλ â„ wλ
where w = wλ(x) is the positive solution to
(5.4)
ââw(x) = λm0w(x)â âaâg(w(x)) x â B+
w|âB+ = 0,
where m0 = infB+ m, âaâ = |a|C(B
+), g(v) = vp0 if 0 †v †1, g(v) = vp1 for v > 1
and p0 = infB+ p, p1 = supB+ p (p0, p1 > 1). Problem (5.4) admits a unique positivesolution for λ > λB+
1 /m0. Using w = AÏB as a subsolution (ÏB the positive eigenfunctionassociated to λB+
1 (ââ), |ÏB|C(B+
)= 1) it follows that
uλ(x0) ℠wλ(x0) ℠A(λ)
where x0 is the center of B+ and A(λ) ⌠(λm0/âaâ)1
p1â1 as λ â â. Thus, we concludelimλââ |uλ|C(Ω) = â.
Let us discuss next the behavior of uλ when say λ â λ2(m) and provided that λ2(m) <â (we are dropping the explicit dependence on m in the notation for short). We arebeginning with the divergence towards infinity in Ωc âȘ Îi where for completeness we aredirectly dealing with the case Îb 6= â . As a first fact, the existence of a uniform subsolutionu for λ2 â ÎŽ2 †λ †λ2 is ensured. In fact, keeping the notation introduced in theconstruction of a subsolution, λΩ
1 (âââ λ2m + aÏΩc) < 0 implies
Ï2 := λΩ1 (âââ λ2m + aÏQ) < 0,
for ÎŽ > 0 small. Using u = ΔÏ2, Ï2 the normalized positive eigenfunction associated to Ï2
we see that the condition for a subsolution is
0 †â(âââ λm + aÏΩc)u = âÏ2 â (λ2 â λ)mu,
in Ωc,
aÏQ\Ωcup(x) †â(âââ λm + aÏΩc)u = âÏ2 â (λ2 â λ)mu + aÏQ\Ωc
u,
MAXIMUM PRINCIPLE 21
in Q \ Ωc and
aup(x) †â(âââ λm + aÏΩc)u = âÏ2 â (λ2 â λ)mu,
in Ω \ Q. Assuming that p > 1 on Î+, if Î+ 6= â , it is clear that such conditions aresatisfied if both Δ > 0 and λ2 â λ are small. If p = 1 at some point of Î+ the subsolutioncan be modified according to the alternative way presented above, getting again a uniformsubsolution in that case.
According to the lines of the corresponding proof in Theorem 7 we show that uλ ââas λ â λ2 uniformly in compacts of Ωc. Thus, let λn â λ2, λ2 â ÎŽ2 †λn < λ2, putun = uλn and observe that u = un solves
(5.5)
ââu(x)â λnm(x)u(x) + au(x) = 0 x â Ωc
u(x) = un(x) x â âΩc.
Setting Un = x â Ω : dist(x,Ωc) < ηn and arguing as in Section 4 a positive sequenceηn â 0 can found such that λUn
1 (âââ λnm + a) = 0 for each n. If Ïn(·, a) stands for thecorresponding associated positive and normalized eigenfunction then Ïn(·, a) â Ï(·, a) inC2,α(Ωc), with Ï the eigenfunction associated to λΩc
1 (âââ λ2m + a).Then, given a compact K â Ωc and M > 0 arbitrary, a small η > 0 can be found so
that K â Ωc,η := x â Ωc : dist(x, âΩc) > η together with
supâΩc,η
MÏn(·, a) †infâΩc,η
u,
for n â„ n0, where u is the uniform subsolution. Since un(x) := MÏn(x, a) is a subsolutionto (5.5), now regarded in Ωc,η, then
un(x) â„ MÏn(x, a) x â Ωc,η.
In particular, for 0 < Ξ < 1 fixed,
un(x) ℠ΞMÏ(x, a) ℠ΞM infK
Ï(·, a) x â K,
for n â„ n1. Since M is arbitrary this means that un ââ uniformly in K.As for the limit behavior of uλ in Îi as λ â λ2 we choose a component, still called Îi,
and introduce strips U+ â Ω+, Uâ â Ωc with small thickness η > 0, U+ = x â Ω+ :dist(x,Îi) < η, Uâ = x â Ωc : dist(x,Îi) > η (see Section 4). Being
un := infÎi
un = un(xn) xn â Îi,
and supposing that un = O(1) as n ââ, we will obtain again a contradiction.We first observe that un †vn in U+ where v = vn is the positive solution to
(5.6)
ââv(x) = λnm(x)v(x)â a(x)(v(x))p(x) in U+
v(x) = un x â Îi
v(x) = un(x) x â âU+ â© âΩ+.
On the other hand un is uniformly bounded in compact subsets of Ω+. In fact, for everyball B â B â Ω+, un †uB where u = uB is the minimal blow-up solution to
ââu = λ2supΩ
m+uâ infΩ
af(u) in B, u|âB = â,
with f(u) = up1 if u †1, f(u) = up0 for u > 1 and p0 = infB p, p1 = supB p. As was shownin Section 4, this gives C2,α uniform bounds for un in every subdomain ΩâČ â ΩâČ â Ω+.
22 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
Now, Schauderâs estimates applied to problem (5.6) ensure us that vn is bounded inC2,α(U+). Finally, since
âun
âÎœ(xn) †âvn
âÎœ(xn),
being Îœ the outer unit normal at Îi, we conclude that âunâÎœ (xn) is bounded above. However,
we use again Theorem 1 to show that such boundedness is not possible. In fact, u = un isthe solution to
(5.7)
ââu(x)â λnm(x)u(x) + au(x) = 0 x â Uâ
u(x) = un(x) x â âUâ.
The fact that λUâ1 (âââ λnm + a) is bounded away from zero implies that the sequence
Ïn, Ï = Ïn being defined as the solution toââÏ(x)â λnm(x)Ï(x) + aÏ(x) = 0 x â Uâ
Ï(x) = 1 x â âUâ,
is bounded in C2,α(Uâ). Taking into account that un â â on âUâ ⩠Ωc, Theorem 1implies that
(5.8) wn(x) :=un(x)Ïn(x)
â„ un(xn) = un x â Uâ.
On the other hand w = wn solves the problemLnw = 0 in Uâ
w = un(·) on âUâ.
where Lnw = ââw â 2âÏn
Ïnâw. Proceeding as in Section 4 we arrive at
limâwn
âÎœ(xn) = â,
Îœ being this time the inner unit normal to Uâ at Îi. The boundedness of Ïn togetherwith its derivatives up to order two implies that âun
âÎœ (xn) ââ and we achieve the desiredcontradiction. Finally, being wn(x) â„ un in U
â we also get that uλ â â uniformly incompacts of Ωc âȘ Îi as λ â λ2.
To finish the proof of Theorem 8 we study the limit behavior of uλ in Ω+ as λ â λ2.Taking λn â λ2 and setting un = uλn the preceding C2,α estimates of un allow us toselect a subsequence unâČ such that unâČ â uâ in C2,α(Ω+). In particular, u = uâ solves theequation in (1.21) for λ = λ2.
On the other hand, assume for completeness that Î+ 6= â . Then, classical Lp estimatesallow us to conclude that unâČ â uâ in C2,α(Ω+ âȘÎ+) and thus uâ = 0 on Î+. Let us showin addition that uâ = â on Îi in the sense that uâ(x) â â as dist(x,Îi) â 0. A firstremark is that for every M > 0, the problem
ââu(x) = λ2m(x)u(x)â a(x)u(x)p(x) x â Ω+
u(x) = M x â Îi
u(x) = 0 x â Î+.
MAXIMUM PRINCIPLE 23
admits a unique positive solution u = uM â C2,α(Ω+) (details are omitted for brevity).Similarly, a unique positive solution u = uM,n â C2,α(Ω+) to
ââu(x) = λnm(x)u(x)â a(x)u(x)p(x) x â Ω+
u(x) = M x â Îi
u(x) = 0 x â Î+,
exists for all n. In addition, Schauderâs estimates imply uM,n â uM in C2,α(Ω+). Nowobserve that un â„ uM,n for n large and thus,
uâ(x) â„ uM (x) x â Ω+,
for all M > 0. This means that uâ â„ u, where u â C2,α(Ω+) is the minimal solution to(1.24), and iv) is proved.
Finally, if m ℠0 in Ω+ then un †un †u in Ω+ (u the minimal solution to (1.24))where u = un is the solution to
ââu(x) = λ2m(x)u(x)â a(x)u(x)p(x) x â Ω+
u(x) = un(x) x â Îi
u(x) = 0 x â Î+.
Therefore, uâ †u and finally uâ = u. The uniqueness in the limit of the subsequence thenimplies that uλ â u as λ â λ2. This finishes the proof of Theorem 8. €
Remarks 5.a) The framework of Theorem 7 allows Î+ = âΩ+ â© âΩ 6= â and moreover that p = 1somewhere on Î+. As was opportunely remarked in the course of the proof, the conclusionsstill hold true under such extreme conditions.
b) The proof of point iv) given in [17] for the case m = 1 (and Ω+ â Ω) can not be carriedover to the case where m is two-signed since the monotonicity of uλ in λ is a crucial elementin the arguments in [17]. In addition, the classical weak maximum principle (1.2)â(1.3)is improperly used in [17] to conclude that un, regarded as a solution to problem (5.7),directly satisfies un(x) â„ un (see Theorem 7.1). To the best of our knowledge, the rightassertion is the alternative estimate (5.8) which is a consequence of Theorem 1. Thesame defect seems to be exhibited in the proof of the corresponding fact in [3] where,nevertheless, (1.2)â(1.3) is still in use due to the superharmonic character of un in Ωc
(Lemma 3.3). However, such approach can not be employed to handle our problem (1.11)where m(x) is a two-signed coefficient.
Our last statement provides an extension to the framework of equation (1.21) of theresults on large solutions contained in Theorems 2 and 3.
Theorem 10. Let Ωc â Ω â RN be C2,α bounded domains, Ωc satisfying (H), m, a, p âCα(Ω), a(x) > 0, p(x) â„ 1 in Ω while
Ωc = x â Ω : p(x) = 1.Then the following properties hold.
i) Problem
(5.9)
ââu(x) = m(x)u(x)â a(x)u(x)p(x) x â Ωu(x) = n x â âΩ,
24 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
admits a positive solution for all n â N if and only if
(5.10) λΩc1 (âââm + a) > 0.
In such case the solution is unique and lies in C2,α(Ω).ii) If Îb = âΩc â© âΩ 6= â then problem
(5.11)
ââu(x) = m(x)u(x)â a(x)u(x)p(x) x â Ωu = â x â âΩ,
does no admit any solution. Moreover, if (5.10) holds and u = un(x) stands for thesolution to (5.9) then
lim un = âuniformly in Ωc. Moreover, un converges in C2,α(Ω+) to the minimal solution u = uâ(x)of the blow-up problem (5.11) when considered in Ω+.
iii) If, on the contrary, Ωc â Ωc â Ω then condition (5.10) is necessary and sufficient forthe existence of a minimal solution u â C2,α(Ω) to (5.11). Such solution is provided bythe limit of un in C2,α(Ω).
Proof. The necessity of condition (5.10) in i) arises from the existence of a positive solutionu â C2(Ω)â©C(Ω) to equation in (5.9). In fact, u defines a strict positive supersolution inΩc and hence the positivity of λΩc
1 (âââm + a) ([16], [5], [6]).Suppose now that (5.10) holds. We are assuming the more adverse conditions to con-
struct a finite positive supersolution u. Namely, Î+ 6= â and p(x) = 1 at some x â Î+.We first extend m, a to Q := B(Ω, ÎŽ) = x â Ω : dist(x,Ω) < ÎŽ, ÎŽ > 0 small, as Cα
functions such that a(x) â„ a0 > 0. In addition, a function p1 â Cα(Q) is chosen such thatp1(x) †p(x) for x â Q, p1 > 1 in Ω+ but p1 = 1 in Q \ Ω. We set in addition:
U = x â RN : dist(x,Î+) < ÎŽand denote U+ = U â©Î©+, Uc = U \U
+. Since the measure of Uc goes to zero with ÎŽ â 0,we can conclude that
λUc1 (âââm + a) > 0.
Therefore, the construction in the proof of Theorem 8 provides the existence of a positivefunction u1 â W 2,q(U) â© C1,ÎČ(U) such that
ââu1 â„ m(x)u1 â a(x)u1p1(x) x â U,
with u1 = 0 on âU . In the same spirit we introduce V = B(Ωc, ÎŽ), V + = V ⩠Ω+,Vc = V \ V
+. For ÎŽ > 0 small we achieve again
λVc1 (âââm + a) > 0,
and get a positive function u2 â W 2,q(V ) â© C1,ÎČ(V ),
ââu2 â„ m(x)u2 â a(x)u2p1(x) x â V,
u2|âV = 0. We finally take
u0(x) =
u1(x) x â Ω, dist(x,Î+) †Ύ2
Ï(x) x â Ω+ÎŽ
u2(x) x â Ω, dist(x,Ωc) †Ύ2 ,
MAXIMUM PRINCIPLE 25
where Ω+ÎŽ = x â Ω+ : dist(x, âΩ+) > ÎŽ
2 and Ï is an smooth extension of both u1 and u2
to Ω+ÎŽ such that Ï > 0 in Ω+
ÎŽ . It is easily checked that u = Mu0 satisfies, for large M â„ 1,
ââu â„ m(x)uâ a(x)up1(x) x â Ω+ÎŽ .
Therefore u defines a strong supersolution to ââu = m(x)u â a(x)up1(x) in Ω and sinceM can be taken so large as to have u â„ 1 we finally conclude
ââu â„ m(x)uâ a(x)up(x) in Ω.
This, in turn, allows to solve (5.9) for all n â N.Let us now show ii) and assume that u â C2(Ω) solves (5.11) with u(x) > 0 for x â Ω.
Then, part i) implies that
λΩc,Ύ
1 (âââm + a) > 0,
Ωc,ÎŽ = x â Ωc : dist(x,Îb) > ÎŽ, for all ÎŽ > 0 small. Thus, by continuity, λΩc1 (âââm +
a) â„ 0. However, λΩc1 (âââm+a) = 0 is ruled out, for taking Ï the positive eigenfunction
associated to λΩc1 (âââm+a) and choosing an arbitrary M â„ 1, a small ÎŽ = ÎŽ(M) exists
such that u â„ MÏ on âΩc,ÎŽ. By comparison we get
u â„ MÏ in Ωc,ÎŽ.
This is not possible due to the arbitrariness of M . Therefore,
λΩc1 (âââm + a) > 0.
This is just (5.10) and in particular implies the existence of the positive solution u = Ï âC2,α(Ωc) to
ââu(x) = m(x)u(x)â a(x)u(x) x â Ωc
u(x) = 0 x â Îi
u(x) = 1 x â Îb.
By comparison in Ωc,ÎŽ and letting ÎŽ â 0+ it follows that u â„ nÏ in Ωc for every n â N,what is not possible.
Regarding the second part of ii) notice that (5.10) allows us to solve (5.9) for all n â N.Our previous argument and comparison show that its solution un satisfies
un(x) â„ nÏ(x) x â Ωc.
This means that un â â uniformly on compacts of Ωc âȘ Îb. The proof of the fact thatun â â uniformly on Îi mimics, via Theorem 1, the corresponding argument given inTheorem 3 (Section 3). Details are therefore omitted for the sake of brevity. The proof ofthe convergence of un towards the minimal solution to ââu = m(x)uâa(x)up(x) in Ω+ isalso skipped for the same reasons (let us point out that it is enough to use the ideas usedin the similar situation in Theorem 8).
To show iii) first observe that the existence of a positive solution to (5.11) directlyimplies (5.10). Conversely, such condition ensures the existence of the solution u = un
to (5.9) for all n â N. To show that un converges to the minimal solution u to (5.11) inC2,α(Ω) it is enough, due to the increasing character of un, to show that un is boundedon compact subsets of Ω. To this end first notice that un †u in Ω\B(Ωc,
ÎŽ2), ÎŽ > 0 small,
where u = u is the minimal solution toââu(x) = m(x)u(x)â a(x)u(x)p(x) x â Ω \B(Ωc,
ÎŽ2)
u = â x â âΩ \B(Ωc,ÎŽ2).
26 J. GARCIA-MELIAN, JULIO D. ROSSI, AND J. C. SABINA DE LIS
In fact, the finiteness of u follows from the fact that p > 1 in Ω \ B(Ωc,ÎŽ2) â Ω+. We
next consider Q = B(Ω, ÎŽ), ÎŽ > 0, and notice that âQ = x â Ω+ : dist(x,Îi) = ÎŽ (recallΩc â Ω in the present case). In addition and by continuity
λQ1 (âââm + aÏΩc) > 0,
for small ÎŽ > 0. Moreover, a smooth function aΔ = aΔ(x) defined in Q can be found suchthat sup aΔ = Ωc, aΔ †aÏΩc and aΔ â a uniformly on compacts of Ωc as Δ â 0. Thus,for small Δ we get
(5.12) λQ1 (âââm + aΔ) > 0.
We now observe that u = un satisfies
ââuâmu + aΔu †ââuâmu + aÏΩcu = âaÏQ\Ωc†0 in Q.
In view of (5.12) we can apply Theorem 1 in the domain Q to get
(5.13) un(x) †supâQ
unÏ(x) x â Q,
with Ï satisfyingââÏ(x)âm(x)Ï(x) + aΔÏ(x) = 0 x â Q
Ï|âQ = 1.
Since âQ â Ω \B(Ωc,ÎŽ2) we have that un is uniformly bounded on âQ and from (5.13) we
achieve both the boundedness of un in Q and in Ω. This finishes the proof. €
Acknowledgements. Supported by Ministerio de Ciencia e Innovacion and FEDERunder grant MTM2008-05824 (Spain) and ANPCyT PICT 03-05009 and UBA X066 (Ar-gentina). J. D. Rossi is a member of CONICET.
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Departamento de Analisis Matematico, Universidad de La Laguna, C/ Astrofısico Fran-cisco Sanchez s/n, 38271 â La Laguna, Spain and, Instituto Universitario de Estudios Avan-zados (IUdEA) en Fısica Atomica, Molecular y Fotonica, Universidad de La Laguna, C/Astrofısico Francisco Sanchez s/n, 38203 â La Laguna , Spain
E-mail address: [email protected]
Departamento de Matematica, FCEyN UBA, Ciudad Universitaria, Pab 1 (1428), BuenosAires, Argentina.
E-mail address: [email protected]
Departamento de Analisis Matematico, Universidad de La Laguna, C/ Astrofısico Fran-cisco Sanchez s/n, 38271 â La Laguna, Spain and, Instituto Universitario de Estudios Avan-zados (IUdEA) en Fısica Atomica, Molecular y Fotonica, Universidad de La Laguna, C/Astrofısico Francisco Sanchez s/n, 38203 â La Laguna , Spain
E-mail address: [email protected]