Terms Existence Solutions Positive Semilinear 1fe.math.kobe-u.ac.jp/FE/FE_pdf_with_bookmark/FE35... · Semilinear Elh.ptic Equationswith Nonlocal Terms By W. ALLEGRETTO A. and BARABANOVA

Funkcialaj Ekvacioj, 40 (1997) 395-409

Existence of Positive Solutions of Semilinear Elh.ptic Equations withNonlocal Terms

By

W. ALLEGRETTO and A. BARABANOVA(University of Alberta, Canada)

0. Introduction

This paper deals with nonlinear partial differential equations which involvea nonlocal term. Our interest is motivated in part by a special equation:

(0. 1) $u_{t}=¥Delta u+u¥psi(x, u, ¥phi(u))$ , $t>0$ , $ x¥in¥Omega$

which has been proposed by Furter and Grinfeld, [11], (see also [6]) to describepopulation dynamics processes. Here $¥Omega$ is a bounded smooth domain in $R^{n}$

and $¥phi(u)$ is a continuous functional representing a nonlocal term. Note that(0.1) without the nonlocal term is well studied (see e.g. [14] and referencestherein), but this is not the case otherwise. In [11] equation (0.1) wasinvestigated in the one-dimensional case and with Neumann Boundary Con-ditions, i.e. no flux was assumed at the boundary, while in [6] a special case of(0.1) was considered with $¥psi(x, ¥phi(u))¥equiv x-¥int_{0}^{1}u(s)ds$ under both Neumann andDirichlet Boundary Conditions. In this paper we consider explicitly equationsof the type (0.1) subject to Dirichlet Boundary Conditions, i.e. there is apopulation reservoir at the boundary, in the case when $u¥psi(x,u, ¥phi(u))$ hassublinear or superlinear growth with respect to $u$ . We suppose explicitly $u=0$

on $¥partial¥Omega$ and $n¥geq 3$ , since if $u=u_{1}(x)$ on $¥partial¥Omega$ with $u_{1}$ nonnegative and nontrivial orif $n=1,2$ then simpler versions of our arguments apply. The change in theboundary conditions from Neumann to Dirichlet creates differences in thesolution behaviour: in particular no longer are there positive constant solutionsto the stationary problem and thus the proof of the existence of a positivestationary solution occupies most of the paper. Stability criteria are thenobtained for some cases of equation (0. 1) and it is shown that the presence of anonlocal term has a stabilizing effect. As far as we are aware, equation (0.1)has not been considered for $n>1$ .

Our main tools are Leray-Schauder Degree Theory, upper-lower solutionprocedures and the Maximum Principle for the proof of existence and Picone’s

Research partly supported by NSERC (Canada).

396 W. ALLEGRETTO and A. BARABANOVA

identity for the proof of uniqueness. We emphasize that upper-lower solutionswork only for some special cases and can not be applied in general. This isrelated to the form of the term $¥psi(x, u, ¥phi(u))$ and especially to the lack ofmonotonicity. We recall in this regard that it was shown by Fukagai, Kusanoand Yoshida, [10], that the general upper-lower solution procedure fails forsuperlinear local equations since upper and lower solutions then turn out to tobe actual solutions, although it is well known that this method works wellfor sublinear local problems. However in the nonlocal case the upper-lowersolution procedure can fail even for sublinear problems as the following ele-mentary equation shows. Consider

(0.2) $-¥Delta u+¥eta¥int_{¥Omega}u=h(x)$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$ ,

with nontrivial $0¥leq h(x)¥in L^{¥infty}(¥Omega)$ . Then results from [3] show that for some$h(x)$ and large $¥eta>0$ the unique solution $u$ must actually be negative somewherein $¥Omega$ . Yet equation (0.2) admits $¥overline{u}=K$ and $¥underline{u}=0$ as an upper-lower solutionpair, with $K$ a large positive constant.

Note also that Brezis and Oswald in [5] used minimization techniques tostudy local sublinear equations. However the lack of a variational structureand the positivity of the nonlocal term do not allow the use of this method inour case.

The paper is structured as follows: we first consider the existence anduniqueness for the sublinear case and then pass to the superlinear problem. Wethen conclude with applications and examples which include equation (0.1), andwhich involve biological models and their stability, and we briefly address thelinear case.

1. The subh.near case

We consider the following nonlocal nonlinear problem in a smoothbounded domain $¥Omega¥subset R^{n}$ , $n¥geq 3$ ,

(1.1) $-¥Delta u+g(x, u)¥phi(u)=f(x, u)$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$

We are interested in positive solutions of (1.1). Here $f(x, u)$ and $g(x, u)$ denotefunctions with nontrivial dependence on $u$ and $¥phi(u)$ is a continuous functionalfrom $H_{0}^{1}$ $(¥Omega)$ to $R$, which maps bounded sets to bounded sets. We suppose herethat $f(x, t)$ is sublinear with respect to $t$ , continuous in $(x, t)$ and reasonablysmooth, that is:

(F1) $¥lim_{t¥rightarrow¥infty}(f(x, t))/t=0$ uniformly for $ x¥in¥Omega$;(F2) $f(x, t)$ is locally Lipschitz continuous with respect to $t$ on $R^{+}$

uniformly for $ x¥in¥Omega$ , where $R^{+}=¥{t ¥in R|t¥geq 0¥}$ ;

Nonlocal Elliptic Equations 397

(F3) $f(x, t)¥geq 0$ for $t>0$, $f(x, t)=f(x,0)$ for $t<0$, $ x¥in¥Omega$ and thereexists $t_{0}>0$ such that $f(x, t)>0$ for $0<t<t_{0}$ .

Note that condition (F1) includes the case when $f(x, t)$ vanishes for large $t$ .The functional $¥emptyset$ represents the nonlocal term in problem (1.1) and we

require that $¥phi(u)$ satisfy

$(¥Phi 1)$ $¥phi(0)=0$ ;$(¥Phi 2)$ $¥phi(u)¥geq¥phi(v)$ for $u$ , $v¥in H_{0}^{1}$ $(¥Omega)$ , $u¥geq v$ in $¥Omega$ with equality holding only

if $u=v$ .

We suppose that $g(x, t)$ is also continuous in $(x, t)$ and locally Lipschitzcontinuous with respect to $t$ uniformly for $ x¥in¥Omega$ and satisfies

(G1) $g(x, t)>0$ for $t>0$, $ x¥in¥Omega$ , $g(x, t)=g(x,0)=0$ for $t¥leq 0$, $ x¥in¥Omega$ and$g(x, t)/t$ is Lipschitz continuous at $t=0$ with respect to $t$ uniformly for $ x¥in¥Omega$ ;

(G2) there exists $ 0<¥beta$ such that $¥varlimsup_{t¥rightarrow¥infty}(g(x, t))/(t^{¥beta})=0$ uniformly for$ x¥in¥Omega$ .

Observe that since we seek positive solutions, the behaviour of $g(x, t)$ and $f(x, t)$

for $t<0$ is irrelevant. If we are interested only in nonnegative nontrivialsolutions of (1.1) then the condition on the smoothness of $g(x, t)/t$ at $t=0$ canbe dropped. We assume finally that the positive solutions of

$-¥Delta u+¥phi(u)g(x, u)=f(x, u)+tw$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$

with $t¥geq 0$ and $0¥leq w¥in C_{0}^{¥infty}$ $(¥overline{¥Omega})$ , are separated from 0. As examples of explicitconditions which suffice for this to hold, we state:

(F4) For $0¥leq u¥in H_{0}^{1}$ $(¥Omega)¥cap L^{¥infty}$ $(¥Omega)$

$¥inf_{x¥in¥Omega}¥frac{f(x,u)}{u}¥geq¥frac{K}{||u||_{L^{¥infty}}^{s}}$

where $K>0$ and $0<s¥leq 1$ and both $K$ and $s$ are independent of the particular$u$ ;

(HI) there exist positive constants $M$ and $l$ $¥geq 0$ such that for any$0<u¥in H_{0}^{1}$ $(¥Omega)$

$||¥frac{¥phi(u)g(x,u)}{u}||_{L^{¥infty}}¥leq M||u||_{L^{¥infty}}^{l}$ .

That these conditions are indeed sufficient will be shown in the sequel. Inaddition we remark that most of the conditions on /, $¥phi$ and $g$ were motivated bythe examples that follow, and were chosen to minimize technicalities in thepresentation. The given proofs almost invariably hold in more general sit-uations with no changes.


First we prove the following lemma.

Lemma 1.1. Let $0<u¥in H_{0}^{1}$ $(¥Omega)¥cap L^{¥infty}$ $(¥Omega)$ solve

(1.2) $-¥Delta u+¥lambda g(x, u)=f(x,u)$ , $¥lambda¥geq 0$

under conditions (G1), (G2), $(¥mathrm{F}1)-(¥mathrm{F}3)$ . then $||u||_{L^{¥infty}}¥leq C$.

Proof. Since $u¥in H_{0}^{1}$ $(¥Omega)$ is positive then $g(x,¥mathrm{w})$ is also positive in $¥Omega$ and wehave that $-¥Delta u¥leq f(x, u)$ . Condition (F1) implies that for any $¥epsilon>0$ there exists$C_{¥epsilon}>0$ such that $f(x,u)¥leq¥epsilon u+C_{¥epsilon}$ . We thus have $-¥Delta u-¥epsilon u¥leq C_{¥epsilon}$ and applying[12, Theorem 8.15] we conclude

$||u||_{L^{¥infty}}¥leq C_{1}(||u||_{L^{2}}+1)¥leq C$

for some constants $C_{1}$ , $C$ independent of $u$ .

Other proofs may also easily be given using bootstrapping arguments andresults in. e.g., [1].

We next establish the existence of positive solutions.

Theorem 1.2. Problem (1.1) has a positive solution $u¥in H_{0}^{1}(¥Omega)¥cap L^{¥infty}(¥Omega)$

under assumptions $(¥mathrm{F}1)-(¥mathrm{F}4)$ , $(¥Phi 1)$ , $(¥Phi 2)$ , (G1), (G2) and (HI).

Proof. We consider a modified problem

(1.3) $-¥Delta u+¥lambda^{+}g(x, u)=f(x, u)$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$ ,

(1.4) $¥lambda=¥phi(u)$

where $¥lambda^{+}$ denotes as usual the positive part of $¥lambda$ . Multiplying first (1.3) by thenegative part $u^{-}$ of any nontrivial solution $u$ we obtain using (G1)

$-¥int_{¥Omega}|Vu^{-}|^{2}=¥int_{¥Omega}f(x,u)u^{-}¥geq 0$

and thus all solutions of (1.3) in $H_{0}^{1}(¥Omega)¥cap L^{¥infty}(¥Omega)$ are nonnegative and $¥lambda=¥lambda^{+}$ .

Since also $g(x, t)/t$ is Lipschitz continuous at $t=0$ with respect to $t$ uniformlyfor $ x¥in¥Omega$ , we rewrite (1.3) as

$-¥Delta u+¥lambda(¥frac{g(x,u)}{u})u¥geq 0$

and $u$ is positive by [12, p. 34].We define the operator $T(¥lambda,u)$ in the following way

$T(¥lambda,u)=¥{¥frac{1}{2}(¥lambda^{+}+¥phi(u)),$ $-¥lambda^{+}(-¥Delta)^{-1}g(x, u)+(-¥Delta)^{-1}f(x, u)¥}$


where $(-¥Delta)^{-1}$ denotes the inverse of the Laplace operator with homogeneousDirichlet boundary conditions. It is easy to see that fixed points of $T$ yieldsolutions of (1.1). We define also the space

$H=R¥times(H_{0}^{1}(¥Omega)¥cap L^{¥infty}(¥Omega))$

with norm

$||(¥lambda, u)||_{H}=(¥lambda^{2}+||u||_{H_{0}^{1}}^{2})^{1/2}+||u||_{L^{¥infty}}$ .

We show first that $T$ acts from $H$ to $H$. Let $T(¥lambda, u)=$ $(¥mu, v)$ . Then it isevident that the $¥mu$ belongs to $R$ . Next, since $u¥in H_{0}^{1}$ $(¥Omega)¥cap L^{¥infty}(¥Omega)$ then by [12]

$||v||_{H_{0}^{1}}¥leq C(||g(x, u)||_{L^{2}}+||f(x, u)||_{L^{2}})$

and (F1) and (G2) imply that

$||v||_{H_{0}^{1}}¥leq C(||u^{¥beta}||_{L^{2}}+||u||_{L^{2}}+1)¥leq C(||u||_{L^{¥infty}}^{¥beta}+||u||_{L^{2}}+1)<¥infty$ .

Finally by [1], [12]

$||v||_{L^{¥infty}}¥leq C(||g(x,u)||_{L^{p}}+||f(x, u)||_{L^{p}})+||v||_{H_{0}^{1}}$

for large $p$ . Thus since $u¥in L^{¥infty}(¥Omega)$ we conclude that $v$ belongs to $ H_{0}^{1}(¥Omega)¥cap$

$L^{¥infty}(¥Omega)$ .

To show that $T$ is continuous consider a sequence $¥{(¥lambda_{n}, u_{n})¥}$ from $H$ suchthat $(¥lambda_{n}, u_{n})¥rightarrow$ $(¥lambda, ¥mathrm{w})$ in the norm of $H$ as $ n¥rightarrow¥infty$ . If we put $T(¥lambda_{n}, u_{n})=$ $(¥mu_{n}, v_{n})$

and $T(¥lambda, u)=$ $(¥mu, v)$ then the continuity of $¥emptyset$ yields that

$|¥mu_{n}-¥mu|¥leq¥frac{1}{2}(|¥lambda_{n}-¥lambda|+|¥phi(u_{n})-¥phi(u)|)¥rightarrow 0$

as $ n¥rightarrow¥infty$ . Moreover

$¥int_{¥Omega}|V(v_{n}-v)|^{2}¥leq¥int_{¥Omega}|¥lambda_{n}||g(x, u_{n})-g(x,u)||v_{n}-v|+|¥lambda_{n}-¥lambda|¥int_{¥Omega}|g(x,u)||v_{n}-v|$

$+¥int_{¥Omega}|f(x, u_{n})-f(x, u)||v_{n}-v|$

$¥leq C(¥int_{¥Omega}|u_{n}-u||v_{n}-v|+|¥lambda_{n}-¥lambda|¥int_{¥Omega}|v_{n}-v|+¥int_{¥Omega}|u_{n}-u||v_{n}-v|)$

$¥leq C||v-v_{n}||_{L^{2}}(||u-u_{n}||_{L^{2}}+|¥lambda-¥lambda_{n}|+||u-u_{n}||_{L^{2}})$ .


Therefore $||v_{n}-v||_{H_{0}^{1}}¥rightarrow 0$ as $ n¥rightarrow¥infty$ . Finally

$||v_{n}-v||_{L^{¥infty}}¥leq C(||¥lambda^{+}g(x,u)-¥lambda_{n}^{+}g(x, u_{n})||_{L^{p}}+||f(x, u) -f(x, u_{n})||_{L^{p}})+||v_{n}-v||_{H_{0}^{1}}$

$¥leq C(|¥lambda_{n}-¥lambda|+||u-u_{n}||_{L^{p}}+||u-u_{n}||_{L^{p}})+||v_{n}-v||_{H_{0}^{1}}$

$¥leq C(|¥lambda_{n}-¥lambda|+||u-u_{n}||_{L^{¥infty}})+||v_{n}-v||_{H_{0}^{1}}¥rightarrow 0$

as $ n¥rightarrow¥infty$ , and we conclude that $T$ is continuous from $H$ to $H$ .Finally we prove that $T$ is a compact operator. Let $¥{(¥lambda_{n}, u_{n})¥}$ be a

bounded sequence from $H$ . By continuity $g(x, u_{n})$ and $f(x, u_{n})$ are bounded in$L^{¥infty}(¥Omega)$ and by [12, Theorem 8.33] $¥{v_{n}¥}$ is bounded in $C^{1+a}(¥overline{¥Omega})$ . Thus we canfind a subsequence of $¥{¥mu_{n}, v_{n})¥}$ which converges in $H$ .

Next, we employ Degree Theory to show the existence of fixed points of$T$ . Note first that all positive solutions (i.e. by (G1) all solutions) of $(¥lambda,u)=$

$vT(¥lambda,u)$ are bounded in $H$ for all $0¥leq v¥leq 1$ . Indeed the equation $(¥lambda, u)=$

$vT(¥lambda, u)$ implies that

(1.5) $-¥Delta u+v¥lambda g(x, u)=vf(x, u)$ ,

(1.6) $¥lambda-¥frac{1}{2}v(¥lambda+¥phi(u))=0$ .

Then multiplying (1.5) by $u$ and integrating over $¥Omega$ yields

$¥int_{¥Omega}|$ Vu $|^{2}¥leq v¥int_{¥Omega}f(x,u)u$

and therefore by (F1)

$||u||_{H_{0}^{1}}^{2}¥leq¥int_{¥Omega}(¥epsilon u^{2}+C_{¥epsilon}u)¥leq¥epsilon||u||_{L^{2}}^{2}+C_{¥epsilon}||u||_{L^{2}}$ .

Choosing $¥epsilon$ small, we obtain that $||u||_{H_{0}^{1}}¥leq C$ . Moreover Lemma 1.1 impliesthat $||u||_{L^{¥infty}}¥leq C$ . Finally (1.6) yields that

$¥lambda(1-¥frac{v}{2})=¥frac{v}{2}¥phi(u)¥leq C$

and we conclude that $¥lambda$ is bounded. A routine homotopy argument impliesthat $¥deg(I-T,B_{R},0)=¥deg(I,B_{R}, 0)=1$ where I is the identity operator and$B_{R}$ denotes a ball in $H$ of large radius $R$ . Here $¥deg$ denotes the Leray-Schauder Degree, [7].

Next, we consider the operator equation

(1.7) $(¥lambda, u)=T(¥lambda, u)+t(0, v)$


with $t¥geq 0$ and $v$ chosen to be the eigenfunction corresponding to the firsteigenvalue $¥mu_{1}$ of $(-¥Delta)$ with homogeneous Dirichlet boundary conditions. Onceagain (G1) implies that all nontrivial solutions of (1.7) are positive. Equation(1.7) is equivalent to the system

(1.8) $-¥Delta u+¥lambda g(x, u)=f(x, u)+t¥mu_{1}v$ ,

(1.9) $¥lambda=¥phi(u)$ .

We want to show that there exists $r>0$ such that any nontrivial solution of(1.8), (1.9) satisfies $||$ $(¥lambda, u)||_{H}>r$.

By [13] we have

$¥mu_{1}+¥sup_{x¥in¥Omega}¥frac{¥phi(u)g(x,u)}{u}¥geq¥inf_{x¥in¥Omega}[¥frac{-¥Delta u+¥phi(u)g(x,u)}{u}]$

$=¥inf_{x¥in¥Omega}[¥frac{f(x,u)}{u}+¥frac{t¥mu_{1}v}{u}]¥geq¥inf_{x¥in¥Omega}[¥frac{f(x,u)}{u}]$

and by (F4) and (HI) there exists $K>0$ such that

$¥frac{K}{||u||_{L^{¥infty}}^{s}}¥leq¥mu_{1}+M||u||_{L^{¥infty}}^{l}$ ,

therefore $||u||_{L^{¥infty}}¥geq C$ . Thus there exists a constant $r>0$ such that $||(¥lambda,u)||_{H}¥geq$

$||u||_{L^{¥infty}}¥geq r$, and so any solution of (1.8), (1.9) with $t¥geq 0$ satisfies $||(¥lambda, u)||_{H}>r$ .

Thus choosing a small $r$ and constructing a suitable homotopy we obtain

$¥deg(I-T, B_{r},0)=¥deg(I-T-t(0, v),B_{r}, 0)=0$ .

The properties of the Leray-Schauder degree, [7], imply that $¥deg(I-T,B_{R}¥backslash ¥overline{B}_{r},0)$

$=1$ and we conclude that the operator $¥mathrm{T}$ has at least one fixed point in theannulus $B_{R}¥backslash ¥overline{B}_{r}$ . We must show that if $(¥lambda, u)$ is one of these fixed points thenboth $¥lambda$ and $u$ are nontrivial. If $¥lambda=0$ then by (1.4) we have $¥phi(u)=0$ andtherefore $u¥equiv 0$ . Thus it follows that $¥lambda$ , and therefore also $¥mathrm{w}$ , is nontrivial andconsequently $u$ is a positive $H_{0}^{1}$ $(¥Omega)¥cap L^{¥infty}$ $(¥Omega)$ -solution of (1.1).

Again by results in [12], it is easy to obtain that $u$ is actually smooth.

We turn next to the investigation of the uniqueness of the solution of(1.1). Assume that the following holds:

(G3) $(g(x, t_{1}))/(t_{1})-(g(x, t_{2})-g(x, t_{1}))/(t_{2}-t_{1})¥leq 0$ for any $0<t_{1}<t_{2}$ .

Theorem 1.3. Let $f(x, u)$, $g(x, u)$ and $¥phi(u)$ satisfy the conditions of The-orem 1.2 and (G3). Assume in addition that for for any $0<t_{1}<t_{2}$


$¥frac{f(x,t_{2})-f(x,t_{1})}{t_{2}-t_{1}}-¥frac{f(x,t_{1})-f(x,0)}{t_{1}}¥leq 0$ .

Then problem (1.1) has unique positive solution from $H_{0}^{1}$ $(¥Omega)¥cap L^{¥infty}$ $(¥Omega)$ .

Proof. We proved already existence and positivity. Suppose that thereexist two positive solutions $u$ , $v$ of problem (1.1), and assume without loss ofgenerality that $¥phi(v)¥geq¥phi(u)$ . Then (1.1) implies

(1.10) $-¥Delta v+g(x, v)¥phi(u)¥leq f(x, v)$ .

Consider now the supplementary problem

(1.11) $-¥Delta w+g(x, w)¥phi(u)=f(x, w)$ in $¥Omega$ , $w=0$ on $¥partial¥Omega$

where $u$ is the solution of (1.1). Inequality (1.10) shows that $v$ is a subsolutionfor problem (1.11). Next (F1), (G1) and (G3) imply that we can find a largeconstant $N$ satisfying $g(x,N)¥phi(u)>f(x,N)$ and $N>$ $¥max¥{||v||_{L_{¥infty}}, ||u||_{L_{¥infty}}¥}$ .

Observe that $N$ is a supersolution for (1.11), and since $f(x, t)$ , $g(x, ¥mathrm{i})$ are locallyLipschitz continuous we can find $M$ $¥geq 0$ such that the operator $f(x, t)$ $-$

$¥phi(u)g(x, t)+Mt$ is monotone on the order interval $¥langle v, N¥rangle$ . We define theoperator $S$ which acts from $H_{0}^{1}$ $(¥Omega)$ to $H_{0}^{1}(¥Omega)$ in the following way

$Sr$ $=p$ $¥Leftrightarrow$ $-¥Delta p+g(x, r)¥phi(u)+Mp=f(x,r)+Mr$ in $¥Omega$ , $p=0$ on $¥partial¥Omega$ .

It is easy to show that $S$ is compact, continuous and monotone on this intervaland maps $(¥mathrm{v},¥mathrm{N})$ into itself. Therefore by the standard upper-lower solutionargument, [7], $S$ has a fixed point $ w¥in¥langle v,N¥rangle$ which is a maximal solution ofproblem (1.11). Furthermore, since $N>||u||_{L_{¥infty}}$ we also have $u¥leq w$ .

We will show that $w¥equiv u$ and to do this we will use Picone’s Identity. Let$¥{u_{n}¥}$ and $¥{w_{n}¥}$ be two sequences of functions from $C_{0}^{¥infty}$ $(¥overline{¥Omega})$ such that $u_{n}¥rightarrow u$ and$w_{n}¥rightarrow w$ in $H_{0}^{1}(¥Omega)$ as $ n¥rightarrow¥infty$ , with $0¥leq w_{n}-u_{n}¥leq w-u$ . Picone’s identity, [2],applied to $u$ and $w_{n}-u_{n}$ gives by direct calculation:

$¥int_{¥Omega}u^{2}[V(¥frac{w_{n}-u_{n}}{u})]^{2}=¥int_{¥Omega}(V(w_{n}-u_{n}))^{2}-¥int_{¥Omega}¥frac{(w_{n}-u_{n})^{2}}{u}(-¥Delta u)$

$=¥int_{¥Omega}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-u_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta u}{u}]$ .

Define $¥Omega_{¥epsilon}=¥{x ¥in¥Omega|w(x)-u(x)>¥epsilon¥}$ for $¥epsilon>0$ . Then we have by the countableadditivity of Lebesgue’s integral

$¥int_{¥Omega}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-u_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta u}{u}]=¥lim_{¥epsilon¥rightarrow 0}¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-u_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta u}{u}]$


and equations (1.1) and (1.11) imply

$¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-u_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta u}{u}]$

$=¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-u_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta(w-u)}{w-u}]$

$+¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w-u)}{w-u}-¥frac{-¥Delta u}{u}]$

$=¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-v_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta(w-u)}{w-u}]$

$+¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{f(x,w)-f(x,u)}{w-u}-¥frac{f(x,u)}{u}]$

$+¥int_{¥Omega_{¥epsilon}}¥phi(u)(w_{n}-u_{n})^{2}[-¥frac{g(x,w)-g(x,u)}{w-u}+¥frac{g(x,u)}{u}]$ .

Since $w>u$ in $¥Omega_{¥epsilon}$ then the conditions of the theorem and (G3) yield that for$x$ $¥in¥Omega_{¥epsilon}$

$¥frac{f(x,w)-f(x,u)}{w-u}-¥frac{f(x,u)-f(x,0)}{u}¥leq 0$

$-¥frac{g(x,w)-g(x,u)}{w-u}+¥frac{g(x,u)}{u}¥leq 0$

and thus

(1.12) $¥int_{¥Omega}u^{2}[V(¥frac{w_{n}-u_{n}}{u})]^{2}¥leq¥lim_{¥epsilon¥rightarrow 0}¥int_{¥Omega_{¥epsilon}}(w_{n}-u_{n})^{2}[¥frac{-¥Delta(w_{n}-u_{n})}{w_{n}-u_{n}}-¥frac{-¥Delta(w-u)}{w-u}]$ .

Passing now to the limit as $ n¥rightarrow¥infty$ we conclude by (1.12) that

$¥int_{¥Omega}u^{2}[V(¥frac{w-u}{u})]^{2}¥leq 0$

and therefore $w¥equiv u$ . But then we also have $u¥geq v$ and $¥phi(u)¥leq¥phi(v)$ which implythat $v¥equiv u$, and the uniqueness follows.

We remark that if $g(x, u)=u^{a}$ and $f(x, u)=u^{¥beta}$ for $u¥geq 0$ with $a$ $>1>$$¥beta>0$ then the uniqueness conditions are satisfied. In this case, $f$ is no longerLipschitz continuous at $u=0$, but it is monotone which suffices for uniqueness.


2. The superh.near case

In this section we study the situation when $f(x, t)$ has a superlinear growthas $ t¥rightarrow¥infty$ . In particular let $f(x, t)$ satisfy the earlier conditions $(¥mathrm{F}2)-(¥mathrm{F}3)$ andinstead of the sublinearity at $¥infty$ we assume that the following holds:

(F5) $¥lim_{t¥rightarrow¥infty}(f(x, t))/(t^{s})=r(x)$ uniformly for $ x¥in¥Omega$ for some $1<s<$$(n+2)/(n-2)$ , $n>2$ and $0<r(x)¥in C^{a}(¥overline{¥Omega})$ .

We consider ffist the case when $f(x, t)$ has the following behaviour near$t=0$

(F6) $f(x, 0)=0$ and $¥lim_{t¥rightarrow+0}(f(x, t))/t<¥mu_{1}$ , where $¥mu_{1}$ is the firsteigenvalue of $(-¥Delta)$ with homogeneous Dirichlet boundary conditions.

We begin by formulating the following local results, which show that if thenonlocal term is small, existence can be proved immediately from the local caseby routine perturbation arguments.

Proposition 2.1. Let $¥tilde{C}=$ { $w|w¥in C^{1}$ $(¥overline{¥Omega})$ , $w=0$ on $¥partial¥Omega$ } equipped with the $C^{1}$

norm and let $f(x, t)$ satisfy (F2), (F3), (F5), (F6). Set $F(u)=$ $(-¥Delta)^{-1}(f(x, u))$ .

Then $F:¥tilde{C}¥rightarrow¥tilde{C}$, $F(u)¥geq 0$ and there exist $r$, $R>0$ such that $¥deg(I-F,B_{R}¥backslash ¥overline{B}_{r}, 0)$

$¥neq 0$ , where $B_{a}$ here denotes the open ball of radius $a$ in $¥tilde{C}$.

Proposition 2.2. There exists $¥delta>0$ such that if $F_{1}$ : $¥tilde{C}¥rightarrow¥tilde{C}$ is a continuousand compact map with $||F_{1}(v)||_{C^{1}}<¥delta$ for $v¥in B_{R}$, $v>0$, and the conclusions ofProposition 2. 1 holdfor $F$, then there exists $0<u¥in B_{R}$ such that $u=F(u)+F_{1}(u)$ .

Except for some minimal changes the proof of these propositions proceeds inthe same way as those given in [4].

Now we state out next theorem.

Theorem 2.3. Consider problem (1.1) under conditions (F2), (F3), (F5),(F6), $(¥Phi 1)$ , $(¥Phi 2)$ , (G1). Let

(2. 1) $||$ $(-¥Delta)^{-1}$ $(¥phi(v)g(x, v))||_{C^{1}}<¥delta$ for $0<v¥in B_{R}$

where $¥delta$ and $R$ are positive constants from Propositions 2.1 and 2.2. Then thereexists a positive solution of problem (1.1).

Proof. The result of the theorem follows immediately from Propositions2.1 and 2.2 if we put $F_{1}v=-(-¥Delta)^{-1}$ $(¥phi(v)g(x, v))$ .

Remark. Note that in many cases condition (2.1) can be easily checked.Indeed if we consider for example $¥phi(v)g(x, v)=¥eta v¥int_{¥Omega}v$ or $¥phi(v)g(x, v)=¥eta¥int_{¥Omega}v$

with $¥eta>0$ then (2.1) holds for small $¥eta$ . Unfortunately we were not able to


prove Theorem 2.3 in the general case i.e. for $¥eta$ large, but we conjecture that thestatement of the theorem is still true.

Next we consider the case when $f(x,0)>0$. We will need the followinglocal result

Proposition 2.4. Let $f(x, t)$ satisfy (F2), (F3), (F5) and $f(x,0)>0$. Thenthe problem

$-¥Delta u=¥lambda f(x, u)$ in $¥Omega$ , $u=0$ on $¥partial¥omega$

has a positive solution in $¥tilde{C}$ for small $¥lambda$ .

This result is well known, see for example [14] and references therein, but wewere unable to find an explicit proof and thus we sketch one for the reader’sconvenience.

Proof. Note first that by classical results [12] the problem $-¥Delta v=f(x, 0)$

has a unique solution in $¥tilde{C}$ such that $v>0$ in $¥Omega$ . So we can find two balls $B_{R}$

and $B_{r}$ in $¥tilde{C}$ such that $v¥in B_{R}¥backslash ¥overline{B}_{r}$ and therefore $¥deg(I-$ $(-¥Delta)^{-1}(f(x, 0))$ ,$B_{R}¥backslash ¥overline{B}_{r},0)=1$ .

We define now $F_{1}w=(-¥Delta)^{-1}[f(x,¥lambda w)-f(x, 0)]$ with $¥lambda>0$ . Then theLipschitz continuity of $f(x, t)$ and estimates from [12] imply that $||F_{1}w||_{C^{1}}¥leq K¥lambda$

for $w¥in B_{R}$ with $K>0$ . Thus if we choose $¥lambda$ small enough we can applyProposition 2.2 and conclude that there exist a positive solution of $-¥Delta w=$

$f(x,¥lambda w)$ in $¥tilde{C}$ . Putting now $u=¥lambda w$ we obtain the statement of the proposition.

We recall that the result of Proposition 2.4 (and consequently what follows)can not hold in general for all $¥lambda>0$ . To see this, choose for example $f(x,u)=$$ u^{s}+¥epsilon$ with $1<s<(n+2)/(n-2)$ and $¥epsilon>0$ and observe that if $v$ is theeigenfunction of $(-¥Delta)$ with Dirichlet boundary conditions corresponding to thefirst eigenvalue then we have:

$¥mu_{1}¥int_{¥Omega}vu=¥int_{¥Omega}v(-¥Delta)u=¥lambda¥int_{¥Omega}v(u^{s}+¥epsilon)$ .

Choosing $¥lambda$ so large that $2¥mu_{1}t¥leq¥lambda(t^{s}+¥epsilon)$ then gives $¥int_{¥Omega}$ $vu=0$, contradicting theassumption $u>0$.

Consider now the parametrized version of (1.1)

(2.2) $-¥Delta u+g(x, u)¥phi(u)=¥lambda f(x, u)$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$ .

Then we can prove the following theorem.

Theorem 2.5. Let $f(x, t)$ satisfy (F2), (F3), (F5) and $f(x,0)>0$, $¥phi(u)$

satisfy $(¥Phi 1)$ , $(¥Phi 2)$ and $g(x, t)$ satisfy (G1). There exist $¥lambda^{*}>0$ such that problem(2.2) with $¥lambda¥leq¥lambda^{*}$ has a positive solution.


Proof. The proof proceeds the same way as in Proposition 2.4 except thatwe put now

$F_{1}w=(-¥Delta)^{-1}[f(x, ¥lambda w)$ $-f(x,0)$ $-¥frac{g(x,¥lambda w)}{¥lambda}¥phi(¥lambda w)]$

and note that (G1) implies that $||g(x,¥lambda w)¥phi(¥lambda w)/¥lambda||_{C}¥leq¥delta/2$ for small $¥lambda$ and$w¥in B_{R}$ .

3. Examples and apph.cations

In this section we discuss applications of the results obtained in theprevious sections. We start by considering equation (1.1) with $g(x, u)=u$ if$u¥geq 0$ and $g(x, u)=0$ if $u<0$, $¥phi(u)=¥int_{¥Omega}u$ and $f(x, u)=(u+¥tau)^{¥gamma}$ where $¥tau¥geq 0$

and $0<¥gamma<1$ , i.e.

(3.1) $-¥Delta u+u¥int_{¥Omega}u=(u+¥tau)^{¥gamma}$ in $¥Omega$ , $u>0$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$ .

In this case $¥phi(u)$ and $g(x, u)$ satisfy hypotheses $(¥Phi 1)$ , $(¥Phi 2)$ , (G1), (G2) and$f(x, u)=(u+¥tau)^{¥gamma}$ satisfies (F1) and (F3). Note that $(u+¥tau)^{¥gamma}$ is not locallyLipschitz continuous on $R^{+}$ but it is $C_{1¥mathrm{o}¥mathrm{c}}^{¥gamma}(R^{+})$ and this smoothness is sufficientfor our needs since as mentioned above, monotonicity can replace smoothness inthe argument. We also note that hypothesis (F4) holds, since

$¥inf_{x¥in¥Omega}¥frac{(u+¥tau)^{¥gamma}}{u}¥geq¥inf_{x¥in¥Omega}¥frac{u^{¥gamma}}{u}¥geq¥frac{1}{||u||_{L^{¥infty}}^{1-¥gamma}}$ .

Finally $||¥int_{¥Omega}u||_{L^{¥infty}}¥leq C||u||_{L^{¥infty}}$ and (HI) holds. Therefore we can apply Theorem1.2 to obtain the existence of a positive solution of (3.1). Moreover since$g(x, ¥mathrm{i})$ is linear and $¥partial f(x, t)/¥partial t$ is decreasing with respect to $t$ for $t>0$ weconclude by Theorem 1.3 that the positive solution of (3.1) is unique.

The remaining examples in this section are inspired by the following modelwhich describes single-species population dynamics with dispersal. The generalapproach to modeling a single species in a domain $¥Omega¥subset R^{n}$ , $n¥geq 3$ , is to supposethat the density of the population $u(x, t)$ satisfies the equation:

(3.2) $u_{t}=¥Delta u+u¥psi(x,u)$ , $t>0$ , $ x¥in¥Omega$

subject to suitable boundary and initial conditions. The function $¥psi$ representsthe “crowding” effect, [15], [11], and is normally supposed to be smooth andsatisfy the condition that there exists $t_{0}>0$ such that $¥psi(x, t_{0})=0$ for $ x¥in¥Omega$ . Itwas suggested in [11] that the “crowding” effect could depend not only on thedensity of the population but also on nonlocal interactions. Mathematically


this situation corresponds to the case when we consider $¥psi=¥psi(x, u, ¥phi(u))$ where$¥emptyset$ is a continuous functional of $u$ . A similar approach can also be found in[6]. In [11] the model given by (3.2) was studied with homogeneous Neumannboundary conditions. In particular, the authors considered the steady statesolutions of (3.2), the bifurcation from constant non-zero solutions and thebehaviour of branches of positive solutions. In what follows we investigatesteady state solutions of (3.2) with homogeneous Dirichlet boundary conditionsand different forms of $¥psi$ .

We start by considering $¥psi=a(1-u)$ $-¥int_{¥Omega}u^{m}$ where $a$, $m$ are positiveconstants such that $a>¥mu_{1}$ and $1¥leq m¥leq 2$ . In the notations of Section 1 wehave that $f(x, u)=au(l - u)$ if $0¥leq u¥leq 1$ and $f(x,u)¥equiv 0$ if $u>1$ , $g(x, u)=u$

and $¥phi(u)=¥int_{¥Omega}u^{m}$ . Then $¥phi(u)$ satisfies $(¥Phi 1)-(¥Phi 2)$ , $f(x,u)$ satisfies $(¥mathrm{F}1)-(¥mathrm{F}3)$ ,

but not (F4). Nevertheless we still can ensure that solutions of (1.7) arebounded away from zero. Indeed, repeating the argument from Theorem 1.2we obtain

$¥mu_{1}+M||u||_{L^{¥infty}}^{m}¥geq a-a||u||_{L^{¥infty}}$

and thus $||u||_{L^{¥infty}}¥geq C$ . Therefore the arguments of Theorems 1.2 and 1.3 holdand we conclude that in this case (3.2) has a unique positive solution $u_{0}$ suchthat $u_{0}¥leq 1$ .

The natural question which arises is that of the stability of $u_{0}$ , and to dealwith this problem we will apply the spectral theory for nonlocal problemselaborated in [9]. Note that it was shown in [8], [9] that in general thelinearized nonlocal problems can have very complicated dynamics, and tosimplify matters we first guarantee that the linearized eigenvalue problem doesnot have complex eigenvalues. We start by writing the linearized operator at$u_{0}$ which is of the form

$Lv$ $=-¥Delta v+bv+mu_{0}¥int_{¥Omega}u_{0}^{m-1}v$

where $b(x)=-a(1-2u_{0}(x))+¥int_{¥Omega}u_{0}^{m}$. If $m=2$ then the operator $L$ is self-adjoint and all eigenvalues of $Lv_{k}=v_{k}v$ are real. If $1¥leq m<2$ then the situa-tion becomes more complicated. We recall that if we let $¥{w_{k}¥}$ be the completesystem of eigenfunction of the corresponding local operator $L_{0}$ , i.e. $L_{0}w_{k}=$

$-¥Delta w_{k}+bw_{k}=¥lambda_{k}w_{k}$, and if $u0$ is such that the following condition holds:

(3.3) $(¥int_{¥Omega}u_{0}w_{k})(¥int_{¥Omega}u_{0}^{m-1}w_{k})$ does not change sign for all $k$

then Lemma 3.9 in [9] states that all eigenvalues of $L$ are real. Unfortunatelycondition (3.3) is very hard to check explicitly except that for $m=2$ it isautomatically satisfied.


Assuming (3.3) holds, then since $w_{1}>0$ and $u_{0}>0$ in $¥Omega$ we conclude that$(¥int_{¥Omega}u_{0}w_{k})$ $(¥int_{¥Omega}u_{0}^{m-1}w_{k})¥geq 0$ for all $k$, and Propositions 3.3. and 3.5 from [9]yield that $v_{k}¥geq¥lambda_{k}$ for all $k$, and moreover $v_{1}>¥lambda_{1}$ . Thus if $¥lambda_{1}¥geq 0$ we concludethat $u_{0}$ is an asymptotically stable solution of the nonlocal problem.

Next we consider the problem (3.2) with $¥psi(x, u)=p(x)u^{¥gamma-1}-¥eta¥int_{¥Omega}u$, i.e.

(3.4) $-¥Delta u+¥eta u¥int_{¥Omega}u=p(x)u^{¥gamma}$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$

where $¥eta$ , $¥gamma$ are positive constants, $1<¥gamma<(n+2)/(n-2)$ and $0<p(x)¥in C^{a}(¥overline{¥Omega})$ .

In this case $f(x, u)=p(x)u^{¥gamma}$ is superlinear and satisfies (F5), (F6), and it followsfrom Theorem 2.3 that problem (3.4) has a positive solution for small $¥eta$ . If wenow put $w=¥lambda u$, then equation (3.4) becomes

$-¥Delta w+¥frac{¥eta}{¥lambda}w¥int_{¥Omega}w=¥lambda^{1-¥gamma}p(x)w^{¥gamma}$ in $¥Omega$ , $w=0$ on $¥partial¥Omega$ .

Thus for any general $¥eta_{0}$ we obtain by choosing $¥lambda=¥eta/¥eta_{0}$ that $w$ solves

$-¥Delta w+¥eta_{0}w¥int_{¥Omega}w=¥lambda^{1-¥gamma}p(x)w^{¥gamma}$ in $¥Omega$ , $w=0$ on $¥partial¥Omega$ .

Note that since $¥eta$ was small it follows that $¥lambda^{1-¥gamma}$ will be large. Therefore weconclude that if we consider the parametrized version of (3.4), i.e. (3.4) withright hand side which is equal to $vp(x)u^{¥gamma}$ then for any $¥eta$ there exists $v_{0}$ such thatfor $v¥geq v_{0}$ our problem will have a positive solution.

Our last example deals with the case when $¥psi(x, u)=a-¥int_{¥Omega}p(x)u(x)dx$ andtherefore we consider the problem

(3.3) $-¥Delta u+u¥int_{¥Omega}p(x)u(x)dx=au$ in $¥Omega$ , $u=0$ on $¥partial¥Omega$ ,

with $0¥leq p(x)¥in C^{a}(¥overline{¥Omega})$ , $p(x)¥not¥equiv 0$ and $a>¥mu_{1}$ where $¥mu_{1}$ is the first eigenvalue of$-¥Delta u=¥mu u$ with homogeneous Dirichlet boundary conditions. The right handside of problem (3.5) is now linear with respect to $u$ and we can not applydirectly results of Sections 1 and 2. But it is possible to check that the firsteigenfunction $0<v_{1}$ of $-¥Delta u=¥mu u$ with $¥int_{¥Omega}p(x)v_{1}(x)dx=a-¥mu_{1}$ solves (3.5).Moreover using the same argument as in Theorem 1.3 we can show that thissolution is unique.

The stability analysis for problem (3.5) shows that the nonlocal term playsa stabilizing role. Indeed, using once again ideas from [9] we write the lin-earized operator at $v_{1}$ in the form

$Lv$ $=-¥Delta v+(¥int_{¥Omega}p(x)v_{1}(x)dx-a)v+v_{1}¥int_{¥Omega}p(x)v(x)dx$

$=-¥Delta v-¥mu_{1}v+v_{1}¥int_{¥Omega}p(x)v(x)dx$ .


Let $¥{v_{k}¥}$ , $ k=1,2¥ldots$ be the complete system of eigenfunction of the operator$-¥Delta v-¥mu_{1}v=¥lambda v$ with corresponding eigenvalues $¥{¥lambda_{k}¥}$ . Then since$(¥int_{¥Omega}v_{1}^{2})$ $(¥int_{¥Omega}pv_{1})>0$ and $(¥int_{¥Omega}v_{1}v_{k})$ $(¥int_{¥Omega}pv_{k})=0$ for all $k>1$ we concludeagain by Lemma 3.9, [9], that all eigenvalues $¥{v_{k}¥}$ of the operator $L$ arereal. Moreover Propositions 3.3 and 3.5, [9], imply that $v_{1}>¥lambda_{1}=0$ and$v_{k}=¥lambda_{k}$ for $k>1$ and once again $v_{1}$ is asymptotically stable.

References

[1] Agmon, S., Douglis, A. and Nirenberg L., Estimates near the boundary for solutions ofelliptic partial differential equations satisfying general boundary conditions, I, Comm. PureAppl. Math., 12 (1959), 623-727.

[2] Allegretto, W., A comparison theorem for nonlinear operators, Ann. Scuola Norm. Sup.Pisa, Cl. Sci., 25 (1971), 41-46.

[3] Allegretto, W. and Barabanova, A., Positivity of solutions of elliptic equations with nonlocalterms, to appear in Proc. Roy. Soc. Edinburgh Sect. A.

[4] Allegretto, W., Nistri, P. and Zecca, P., Positive solutions of elliptic non-positone problems,Diff. and Int. Eqns., 5 (1992), 95-101.

[5] Brezis, H. and Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Ana.,Theory, Methods & Applications, 10 (1986), 55-64.

[6] Calsina, A. and Perello, C., Equations for biological evolution, to appear in Proc. Roy.Soc. Edinburgh Sect. A.

[7] Deimling, K., Nonlinear functional analysis, Springer-Verlag, New York, 1985.[8] Fiedler, B. and Polacik, P., Complicated dynamics of a scalar reaction-diffusion equation

with a nonlocal term, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 167-192.[9] Freitas, P., A nonlocal Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburgh

Sect. A, 124 (1994), 169-188.[10] Fukagai, N., Kusano, T. and Yoshida, K., Some remarks on the supersolution-subsolution

method for superlinear elliptic equations, J. Math. Anal. Appl., 123 (1987), 131-141.[11] Furter, J. and Grinfeld, M., Local vs. non-local interactions in population dynamics, J.

Math. Biol., 27 (1989), 65-80.[12] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of the second order,

Springer-Verlag, Berlin, 1983.[13] Krasnosel’skii, M. A., Positive solutions of operator equations, Noordhoff, Groningen, 1964.[14] Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM

Rev., 24 (1982), 441-467.[15] Okubo, A., Diffusion and ecological problems. Mathematical models, Springer-Verlag, Berlin,

1980.

nuna adreso:

Department of Mathematical SciencesUniversity of AlbertaEdmonton, AlbertaCanada

(Ricevita la 31-an de majo, 1996)

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Terms Existence Solutions Positive Semilinear 1fe.math.kobe-u.ac.jp/FE/FE_pdf_with_bookmark/FE35... · Semilinear Elh.ptic Equationswith Nonlocal Terms By W. ALLEGRETTO A. and BARABANOVA