Embed Size (px)
Citation preview
Calc. Var.DOI 10.1007/s00526-013-0698-1 Calculus of Variations
Biharmonic elliptic problems involving the 2nd Hessianoperator
Fausto Ferrari · Maria Medina · Ireneo Peral
Received: 31 July 2013 / Accepted: 16 December 2013© Springer-Verlag Berlin Heidelberg 2013
Abstract In this paper we will study the equation
�2u = S2(D2u), � ⊂ R
N ,
with N = 3, where S2(D2u)(x) = ∑1≤i< j≤N λi (x)λ j (x), being λi , the solutions to the
equation
det(λI − D2u(x)
) = 0,
i = 1, . . . , N , and � is a bounded domain with smooth boundary. We deal with severalboundary conditions looking for the appropriate framework to get existence and multiplicityof nontrivial solutions. This kind of equation is related to some models of growth, and forthis reason it is natural to study the effect of zero order local reaction terms of the typeFλ(x, u) = λ|u|p−1u, with λ ∈ R, λ > 0, and 0 < p < ∞, and also the solvability of theboundary problems with a source term f satisfying some integrability hypotheses.
Communicated by P. Rabinowitz.
F. Ferrari was partially supported by the GNAMPA project: “Equazioni non lineari su varietà: proprietàqualitative e classificazione delle soluzioni” and FP7-IDEAS-ERC Starting Grant 2011 #277749(EPSILON). M. Medina and I. Peral are supported by project MTM2010-18218 of MICINN, Spain.
F. FerrariDipartimento di Matematica, dell’Università di Bologna,Piazza di Porta S. Donato, 5, 40126 Bologna, Italye-mail: [email protected]
M. Medina · I. Peral (B)Departamento de Matemáticas, Universidad Autónoma de Madrid,Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spaine-mail: [email protected]
M. Medinae-mail: [email protected]
123
F. Ferrari et al.
Mathematics Subject Classification (2010) 35J50 · 35J60 · 35J62 · 35J96 · 35G20 ·35G30
1 Introduction
We consider the following model problem{�2u = S2(D2u), � ⊂ R
N ,
B(u) = 0, ∂�,(1.1)
where S2(D2u)(x) = ∑1≤i< j≤N λi (x)λ j (x), being λi , i = 1, . . . , N , the solutions to the
equation
det(λI − D2u(x)
) = 0,
�2 the bi-Laplacian operator, � a bounded domain with smooth boundary and N = 3. Theconditions on� hold throughout the paper and will not be in general specified in what follows.We denote by B(u) some boundary conditions (mainly Dirichlet and Navier conditions).
The case N = 2 appears as the stationary part of a model of epitaxial growth of crystals(see [11,19]) initially studied in [12]. In dimension N = 3 the model can be seen as thestationary part of a 3-dimensional growth problem driven by the scalar curvature.
We find natural restrictions in the study of the boundary problems, linked to the followingsubjects.
(1) The deep dependence on the boundary conditions of the variational formulation of thenonlinear term S2(D2u).
(2) The use of the critical point theory and the dimension N (conditioned by some compact-ness properties at the gradient level).
(3) When the variational formulation is not valid (Navier conditions), we need some alter-native techniques that also depend on some reactions and the source terms.
It is natural, both from the theoretical and applied point of view, to study the effect of asource term. Indeed, that term describes, roughly saying, the amount of material providedto the system from the exterior, and moreover, takes in account of a local reaction termin the behavior of the problem. In the case of Dirichlet boundary conditions a reactionterm influences the multiplicity of nontrivial solutions. Moreover, sometimes, it changes thestability of the equilibrium to the zero solution. In the case of Navier boundary conditions,it is worth to point out that, when the problem does not admit a variational formulation, asuitable reaction term provides a nontrivial solution.
We list our main results.In what follows, suppose that � ⊂ R
3 is a bounded smooth set.Let us consider the following Dirichlet problem
⎧⎪⎨
⎪⎩
�2u = S2(D2u), �,
u = 0, ∂�,∂u
∂n= 0, ∂�.
(1.2)
Theorem 1.1 Problem (1.2) has at least one nontrivial solution.
Consider now the problem with a reaction term with Navier conditions:
123
Biharmonic elliptic problems
⎧⎨
⎩
�2u = S2(D2u)+ λu, �,
u = 0, ∂�,
�u = 0, ∂�.
(1.3)
Recalling the bifurcation theory we achieve information about the solvability of this problem.Indeed, in Sect. 4 we prove the following result by using the classical global bifurcationtheorem by Rabinowitz. See [21].
Theorem 1.2 Let λ1 be the first eigenvalue of �2 in � with Navier boundary condi-tions, which is simple. Then, there exists an unbounded continuum of pairs (λ, u), u ∈W 2,2(�)∩ W 1,2
0 (�) branching-off from (λ1, 0), where every u is a solution to (1.3) with thecorresponding λ.
Moreover, by means of Fixed Point methods, we can study the solvability of the problemwith Navier conditions and a source term, that is,
⎧⎨
⎩
�2u = S2(D2u)+ μ f (x) x ∈ �,u = 0 x ∈ ∂�,�u = 0 x ∈ ∂�,
(1.4)
where μ > 0. Hence, we can obtain the next result.
Theorem 1.3 Let f ∈ L1(�). Then, there exists μ0 > 0 such that for every μ satisfying0 < μ < μ0, there exists u ∈ W 1,2
0 (�) ∩ W 2,2(�) solution to problem (1.4).
In order to point out other results contained in this paper we describe how we organizedit. More precisely, Sect. 2 is devoted to analyze, when it is possible, a variational formulationof the problem. We will need to use some highly nontrivial results collected in [6]. As aconsequence we obtained that the variational formulation is possible for Dirichlet boundaryconditions while it is not possible for Navier conditions.
In Sect. 3 we study the existence of a nontrivial solution to the Dirichlet problem (1.2)essentially by using critical points theory. Next we study the influence of a reaction term ofthe form |u|p−1u in the multiplicity of solutions, see Theorem 3.3.
Section 4 deals with the analysis of the problem with Navier conditions (prescribing u = 0and �u = 0 on the boundary), where a variational formulation is not admitted. We solvethe problem with a reaction term λu by bifurcation arguments. Next, in Sect. 5 we facethe problem with Navier conditions (it can be reproduced in the same way for Dirichletconditions) and a source term. An important remark explaining the limitation of the prooffor higher dimensions is stated at the end of this section.
Eventually, in the last section, we collect some open problems that seem to be interestingfrom the theoretical and the applications point of view.
2 The 2-Hessian: functional setting and variational formulation
In this section we expose some highly nontrivial results concerning the regularity of thenonlinear term S2(D2u). Let S2 be the 2-th Hessian operator. Namely we set
S2(D2u)(x) =
∑
1≤i< j≤N
λi (x)λ j (x),
where λi (x) i = 1, . . . , N is the i-th eigenvalue of the symmetric matrix D2u(x).
123
F. Ferrari et al.
For further details about k-th Hessian operators, see [27] and, for some applications seealso [13–15].
We remind moreover that S2 can also be written in the following way:
S2(D2u(x)) =
∑
1≤i< j≤N
det(D2i j u(x)),
where
det(D2i j u(x)) = ∂i i u(x)∂ j j u(x)− (∂i j u(x))
2.
On the other hand we remark that for u conveniently smooth,
∂i j (∂i u∂ j u)− 1
2∂i i ((∂ j u)
2)− 1
2∂ j j ((∂i u)
2)
= ∂i (∂ j i u∂ j u + ∂i u∂ j j u)− ∂i (∂ j u∂i j u)− ∂ j (∂i u∂ j i u)
= ∂i i u∂ j j u − ∂i j u∂i j u = ∂i i u∂ j j u − (∂i j u)2. (2.1)
As a consequence
S2(D2u(x)) =
∑
1≤i< j≤N
(∂i i u(x)∂ j j u(x)− (∂i j u(x))
2)
=∑
1≤i< j≤N
(
∂i j (∂i u∂ j u)− 1
2∂i i ((∂ j u)
2)− 1
2∂ j j ((∂i u)
2)
)
. (2.2)
Moreover
(�u(x))2 =(
N∑
i=1
∂i i u(x)
)2
=N∑
i=1
(∂i i u(x))2 + 2
∑
1≤i< j≤N
∂i i u(x)∂ j j u(x),
so that
(�u(x))2 −N∑
i=1
(∂i i u(x))2 = 2
∑
1≤i< j≤N
∂i i u(x)∂ j j u(x)
and
(�u(x))2 −N∑
i=1
(∂i i u(x))2 − 2
∑
1≤i< j≤N
(∂i j u(x))2
= 2∑
1≤i< j≤N
∂i i u(x)∂ j j u(x)− 2∑
1≤i< j≤N
(∂i j u(x))2.
Thus
(�u(x))2 −N∑
i, j=1
(∂i j u(x))2 = 2
∑
1≤i< j≤N
(∂i i u(x)∂ j j u(x)− (∂i j u(x))2),
that is
(�u(x))2− | D2u(x) |22 = 2∑
1≤i< j≤N
(∂i i u(x)∂ j j u(x)− (∂i j u(x))2)
= 2∑
1≤i< j≤N
det(D2i j u(x)) = 2S2(D
2u(x)).
123
Biharmonic elliptic problems
For the reader convenience we recall the definition of the Hardy space H1(RN ). Let F bethe Fourier transform in R
N . We define R j , the classical Riesz transform, as
R j = ∂
∂x j(−�)− 1
2 , j = 1, 2, . . . , N ,
or equivalently,
R j = F−1ix j
|x |F, j = 1, 2, . . . , N .
Consider also
ht (x) = 1
t Nh
( x
t
), where h ∈ C∞
0 (RN ), h(x) ≥ 0 and
∫
RN
h dx = 1.
Definition 2.1 The Hardy space H1(RN ) is defined in an equivalent way as follows
H1(RN ) = { f ∈ L1(RN ) | R j ( f ) ∈ L1(RN ), j = 1, 2, . . . , N }= { f ∈ L1(RN ) | sup | f ∗ ht (x)| ∈ L1(RN )},
See [23,24] for further details. The following particular case of a result by Coifman et al.(see [6]) gives a distributional sense to the identities above for functions in W 2,2(RN ).
Lemma 2.2 (Coifman, Lions, Meyer and Semmes) Let U, V vector fields in RN such that
∇U, ∇V ∈ [L2(RN )
]N×Nand div (U ) = div (V ) = 0 in D′(RN ). Then
N∑
i, j=1
∂i j (Ui Vj ) ∈ H1(RN ),
where Ui and Vj denote the i th and jth components of U and V respectively.
The proof of Lemma 2.2 involves some techniques from Harmonic Analysis and an adap-tation of the ideas by Luc Tartar on compensation compactness. See, for the last subject, thereferences [25,26]. We will use a localization of the following result, which is a Corollary ofLemma 2.2.
Lemma 2.3 If u ∈ W 2,2(RN ), N ≥ 2 then
∑
i �= j
(
∂i j (∂i u∂ j u)− 1
2∂i i ((∂ j u)
2)− 1
2∂ j j ((∂i u)
2)
)
∈ H1(RN ).
Proof For any 1 ≤ i < j ≤ N we define
Ui, j = (Ui, j(1) ,U
i, j(2) , . . . ,U
i, j(N ))
where Ui, j(k) = 0, if k �= i, j and Ui, j
(i) = −∂ j u, Ui, j( j) = ∂i u. In particular
div(Ui, j ) = 0.
Now let us denote U = Ui, j and V = Ui, j . Then Ui Vi = (∂ j u)2, U j Vj = (∂i u)2, andUi Vj = U j Vi =− ∂ j u∂i u, otherwise Uk Vl = 0, whenever k �= i, j or l �= i, j. In particularrecalling Lemma 2.2 it results
−1
2
N∑
k,l=1
∂kl(Uk Vl) = ∂i j (∂i u∂ j u)− 1
2∂i i ((∂ j u)
2)− 1
2∂ j j ((∂i u)
2) ∈ H1(RN ).
123
F. Ferrari et al.
In particular, by linearity, we conclude
∑
i< j
(
∂i j (∂i u∂ j u)− 1
2∂i i ((∂ j u)
2)− 1
2∂ j j ((∂i u)
2)
)
∈ H1(RN )
� By using Lemma 2.3, integrating by parts we get, for every v ∈ C∞
0 (�),∫
�
(�2u − S2(D2u))v =
∫
�
�u�v
−∫
�
∑
1≤i< j≤N
((∂i u∂ j u)∂i jv + 1
2∂i (∂ j u)
2∂iv + 1
2∂ j (∂i u)
2∂ jv)
=∫
�
�u�v −∫
�
∑
1≤i< j≤N
(∂i u∂ j u∂i jv + ∂ j u∂i j u∂iv + ∂i u∂ j i u∂ jv
). (2.3)
Hence, if we consider the functional
G(u) = 1
2
∫
�
|�u|2 −∫
�
∑
1≤i< j≤N
∂i j u∂i u∂ j u,
defined in W 2,20 (�), then
dG(u + εv)
dε
∣∣∣∣ε=0
=∫
�
�u�v −∫
�
∑
1≤i< j≤N
(∂i u∂ j u∂i jv + ∂ j u∂i j u∂iv + ∂i u∂ j i u∂ jv
).
We can summarize the previous calculation in the following result.
Proposition 2.4 Consider the functional
G(u) = 1
2
∫
�
|�u|2 −∫
�
∑
1≤i< j≤N
∂i j u∂i u∂ j u, (2.4)
with u ∈ W 2,20 (�). Then, the critical points of G are weak solutions to the Dirichlet problem
(1.2).
Remark 2.5 It is easy to check that if we consider Navier boundary conditions, that is,prescribing u = 0 and �u = 0 on the boundary, we find some integral terms on ∂� whichdo not vanish.
That means that the Navier conditions do not admit such a Lagrangian without boundaryintegrals to have a variational formulation.
The fourth derivatives of the fundamental solution of biharmonicequations. By the classical Malgrange-Ehrenpreis Theorem we know that�2 has a funda-mental solution, that is, the equation
�2 E(x) = δ0
has a distributional solution (see [9,10,18]). By applying the Fourier transform in S ′(RN )
we obtain that
�2u = δ0 ≡ |ξ |4F(u)(ξ) = 1.
123
Biharmonic elliptic problems
The homogeneity of the �2 operator allows to find an explicit form for E in terms of thedimension. See [17], formula (4.61) at page 121. Therefore, if we take f ∈ L1(RN ) and,for instance, such that f ≡ 0 in the complementary of a compact set, then a solution to�2u = f , can be obtained by the convolution,
u(x) =∫
RN
f (y)E(x − y)dy.
It is easy to check (in fact it is a classical computation) that any fourth order derivative,
D4i, j,k,l E(x) = Ki, j,k,l(x)
|x |N,
where for all r > 0, Ki, j,k,l(r x) = Ki, j,k,l(x), that is, K is homogeneous of zero degree and∫
SN−1
Ki, j,k,l(x̄)dx̄ = 0,
where SN−1 = {x ∈ RN : ‖x‖ = 1}.Therefore any Ki, j,k,l is a classical Calderon–Zygmund
kernel, and thus the operator defined on integrable functions by
Ti, j,k,l( f )(x) = limε→0
∫
ε<|x−y|< 1ε
Ki, j,k,l(x)
|x |Nf (x − y)dy
verifies that if 1 < p < ∞, there exists C p > 0 such that
||T f ||p ≤ C p|| f ||p.
See [1,2,5,22].For the extremes of the interval the result is false, however we have for p = 1 that there
exists a C > 0 such that,
|{x ∈ RN | |T f (x)| > λ}| ≤ C
λ
∫
{x∈RN | | f (x)|>λ}f (x)dx .
Just to solve this anomaly the Hardy space has been defined. More precisely, if f ∈ H1(RN ),
u(x) =∫
RN
f (y)E(x − y)dy ∈ W 4,1(RN ).
So by Lemma 2.3 u ∈ W 2,2(R
N)
and g ∈ L2(R
N)
results in S2(D2u
) + g ∈ H1(R
N)
and for �2u ∈ H1(R
N)
one finds in the previous lines that u ∈ W 4,1(R
N).
Moreover, as a consequence of the Lemma 2.3 and the previous remarks we have thefollowing result.
Proposition 2.6 Let u ∈ W 2,2(RN ) be a solution to
�2u = S2(D2u)+ f,
in RN where f ∈ L2(RN ), f (x) = 0 in R
N \BR(0). Then, u ∈ W 4,1(RN ).
123
F. Ferrari et al.
3 The Dirichlet problem
Consider the problem (1.2) in a bounded smooth set � ⊂ R3.
By Proposition 2.4, we know that critical points of G are solutions of the problem (1.2).Therefore, the purpose now is to study the behavior of this functional respect to the mountainpass theorem. See [4].
Indeed, we can prove that G satisfies the Palais–Smale condition.
Lemma 3.1 Assume {un}n∈N ⊂ W 2,20 (�) is a bounded Palais–Smale sequence for G, that
is, {un}n∈N verifies
(1) ||�un ||L2(�) ≤ C,(2) G(un) → c as n → ∞,(3) G′(un) → 0 in W −2,2(�).
Then there exists a subsequence of {un}n∈N that converges in W 2,20 (�).
Proof Since ||�un ||L2(�) ≤ C , up to a subsequence, by the Rellich–Kondrachov Theorem
for the space W 2,20 (�) (see for example [17]) we know that
un ⇀ u in W 2,20 (�),
un → u in L p(�), for all 1 ≤ p < ∞,
un → u uniformly in �.
Moreover, ||∇un ||W 1,20 (�)
≤ C . Hence, applying now the Rellich–Kondrachov Theorem
for W 1,20 (�) we also obtain that
∇un → ∇u in [L p(�)]N , 1 ≤ p < 6.
On the other hand, we can write condition (3) as
�2un = S2(D2un)+ yn, un ∈ W 2,2
0 (�) and yn → 0 in W −2,2(�).
Multiplying here by (un − u), we have∫
�
�un�(un − u) dx =∫
�
(un − u)S2(D2un) dx +
∫
�
yn(un − u) dx . (3.1)
The last term on the right hand side goes to zero due to the convergence of yn . For thefirst one, integrating by parts, we obtain
∫
�
(un − u)S2(D2un) dx =
∫
�
∑
1≤i< j≤N
(∂i un∂ j un∂i j (un − u)
+∂ j un∂i j un∂i (un − u)+ ∂i un∂ j i un∂ j (un − u))
=∫
�
∑
1≤i< j≤N
(− ∂i i un∂ j un∂ j (un − u)− ∂i un∂i j un∂ j (un − u)
+∂ j un∂i j un∂i (un − u)+ ∂i un∂ j i un∂ j (un − u))
=∫
�
∑
1≤i< j≤N
(− ∂i i un∂ j un∂ j (un − u)+ ∂ j un∂i j un∂i (un − u)
).
123
Biharmonic elliptic problems
Using now the Hölder inequality, the boundedness of un in W 2,20 (�) and the L p conver-
gences, we can conclude∫
�
(un − u)S2(D2un) dx ≤ C
∫
�
|D2un ||∇un ||∇(un − u)| dx
≤ C ||�un ||L2(�)||∇un ||L4(�)||∇un − ∇u||L4(�) → 0. (3.2)
Therefore,∫
�
�un�(un − u) dx → 0 when n → ∞.
Otherwise, by the weak convergence in W 2,20 (�),
∫
�
�u�(un − u) dx → 0 when n → ∞,
and hence, subtracting these two terms we reach∫
�
|�(un − u)|2 dx → 0,
and therefore Palais–Smale condition is satisfied. � Remark 3.2 This result does not hold for N = 4. The convergence of the gradients in L4
cannot be obtained by the Rellich–Kondrachov Theorem, because the critical exponent atthe level of the gradients is
2N
N − 2= 4.
Therefore, without any other argument, we cannot pass to the limit in (3.2).
On the other hand, just by the Hölder and Sobolev inequalities we can obtain informationabout the geometry of the functional. In fact we can see that
G(u) ≥ 1
2
∫
�
|�u|2 dx − C
⎛
⎝∫
�
|�u|2 dx
⎞
⎠
12⎛
⎝∫
�
|∇u|4 dx
⎞
⎠
12
≥ 1
2
∫
�
|�u|2 dx − c1
⎛
⎝∫
�
|�u|2 dx
⎞
⎠
32
= 1
2||�u||2L2(�)
− c1||�u||3L2(�)
≡ h(||�u||L2(�)
),
where
h(s) = 1
2s2 − c1s3. (3.3)
Hence, we can prove the Theorem 1.1
123
F. Ferrari et al.
Proof of Theorem 1.1 We proceed in several steps.
Step 1. It is easy to check that there exists a function ψ ∈ W 2,20 (�) such that
∫
�
∑
1≤i< j≤N
∂i jψ∂iψ∂ jψdx > 0.
As a consequence, G(sψ) < 0 for s large enough.
Step 2. According to the lower estimates, it can be easily checked that h, defined by (3.3),has a local positive maximum. We will prove that this fact implies that the functional has amountain pass critical point.
Step 3. G has the geometry of the mountain pass lemma.Recalling together Step 1 and Step 2, we can conclude that there exist α, β > 0 such that
the following conditions hold:
(1) G(u) ≥ β for all u ∈ W 2,20 (�) with ‖�u‖L2(�) = α.
(2) There exists v ∈ W 2,20 (�), v > 0 such that ‖�v‖L2(�) > α and G(v) < β.
That is, G has the mountain pass geometry. See [4]. Define now
= {γ ∈ C([0, 1],W 2,2
0 (�))
| γ (0) = 0, γ (1) = v},and the mini-max value
c = infγ∈ max
t∈[0,1] G[γ (t)].
Applying the Ekeland variational principle (see [8]), there exists a Palais–Smale sequence tothe level c, i.e., there exists {un}n∈N ⊂ W 2,2
0 (�) such that
(1) G(un) → c as n → ∞,(2) G′(un) → 0 in W −2,2(�).
Step 4. If {un}n∈N ⊂ W 2,20 (�) is a Palais–Smale sequence for G at the level c, then there
exists C > 0 such that ||�un ||L2(�) < C .
If u ∈ W 2,20 (�), using the equivalent form (2.2) and integrating by parts we find that
∫
�
u S2(D2u) dx = 3
∫
�
∑
1≤i< j≤N
∂i u∂ j u∂i j u. dx . (3.4)
Then if {un}n∈N ⊂ W 2,20 (�) is a Palais–Smale sequence for G at the level c, by calling
〈yn, un〉 = 〈G′(un), un〉,c + o(1) = G(un)− 1
3〈G′(un), un〉 + 1
3〈yn, un〉
≥(
1
2− 1
3
) ∫
�
|�un |2 dx − 1
3||yn ||W−2,2(�)
⎛
⎝∫
�
|�un |2 dx
⎞
⎠
12
=(
1
2− 1
3
)
‖�un‖2L2(�)
− 1
3||yn ||W−2,2(�‖�un‖L2(�).
Then we easily conclude.
Step 5. By using Lemma 3.1, G satisfies the Palais–Smale condition at the level c. Therefore
123
Biharmonic elliptic problems
(1) G(u) = limn→∞ G(un) = c.(2) G′(u) = 0, thus
�2u = S2(D2u), u ∈ W 2,2
0 (�).
In other words, u is a mountain pass type solution to the problem (1.2). � 3.1 Influence of a reaction term in the multiplicity of solutions
Consider now the problem
⎧⎪⎨
⎪⎩
�2u = S2(D2u)+ λ|u|p−1u x ∈ �,u = 0 x ∈ ∂�,∂u
∂n= 0 x ∈ ∂�,
(3.5)
where the hypotheses assumed over the problem (1.2) hold, and we suppose λ > 0 and0 < p < ∞. The idea now is to analyze how the reaction term affects the solvability ofthe problem. We will see that indeed this influence will depend on the power p, obtainingdifferent results for the three cases p < 1, p = 1 and p > 1. First of all, we recall that thisproblem keeps having a variational formulation and, analogously to the previous case, it canbe checked that the critical points of the functional
Gλ : W 2,20 (�) → R
defined by
Gλ(u) = 1
2
∫
�
|�u|2 −∫
�
∑
1≤i< j≤N
∂i j u∂i u∂ j u − λ
p + 1
∫
�
|u|p+1
give us exactly the weak solutions of problem (3.5).Since for N = 3 the critical exponent in the Rellich–Kondrachov Theorem is +∞, it can
be easily proved that Gλ satisfies the Palais–Smale condition for the whole range of p in avery similar way to the proof of Lemma 3.1. Concerning to the geometry, in this case theradial minorant will be
Gλ(u) ≥ 1
2
∫
�
|�u|2 dx − C
⎛
⎝∫
�
|�u|2 dx
⎞
⎠
12⎛
⎝∫
�
|∇u|4 dx
⎞
⎠
12
− λ
p + 1
∫
�
|u|p+1
≥ 1
2||�u||2L2(�)
− c1||�u||3L2(�)− λc2||�u||p+1
L2(�)
≡ g(||�u||L2(�)
),
where
g(s) = 1
2s2 − c1s3 − λc2s p+1. (3.6)
It is clear that the geometry of the functional depends on p and motivates how the behaviorabout existence and multiplicity is:
123
F. Ferrari et al.
We can state now the following theorem,
Theorem 3.3 Let 0 < p < ∞.
(i) If p < 1 there exists a λ0 > 0 such that if 0 < λ < λ0, problem (3.5) has at least twonontrivial solutions.
(ii) If p > 1 problem (3.5) has at least one nontrivial solution for every λ ≥ 0.(iii) If p = 1 problem (3.5) has at least one nontrivial solution whenever 0 < λ < λ1,
where λ1 denotes the first eigenvalue of �2 in � with Dirichlet boundary conditions.
Proof We split the proof in the three cases:
(i) 0 < p < 1: As in the proof of Theorem 1.1, we proceed in several steps:
Step 1. It is easy to check that:
(a) There exists a function φ ∈ W 2,20 (�) such that
∫
�
|φ|p+1 dx > 0.
(b) There exists a function ψ ∈ W 2,20 (�) such that
∫
�
∑
1≤i< j≤N
∂i jψ∂iψ∂ jψdx > 0.
As a consequence
Gλ(tφ) < 0 for t small enough and Gλ(sψ) < 0 for s large enough.
Step 2. It is easy to check that for 0 < λ < λ0 small enough, g, defined by (3.6),has a local negative minimum and a local positive maximum. Hence, we will searchfor a local minimum and a mountain pass critical point of the functional.Step 3. There exists λ0 such that if 0 < λ < λ0, then Gλ has a local minimum u0,with Gλ(u0) < 0We follow here the ideas of [16]. Take λ0 such that g attains its positive maximumat rmax > 0, and denote by r0 the lower positive zero of g. Fix r1 such that r0 <
r1 < rmax , with g(r1) > 0. Define now a nonincreasing cutoff function τ ∈ C∞,verifying
τ : R+ → [0, 1],{τ(s) = 1, if s ≤ r0
τ(s) = 0, if s ≥ r1.
123
Biharmonic elliptic problems
Consider now the truncated functional
Gλ,τ (u) = 1
2
∫
�
|�u|2 − τ(‖�u‖L2(�))
∫
�
∑
1≤i< j≤N
∂i j u∂i u∂ j u − λ
∫
�
|u|p+1.
This functional clearly has a local minimum u0 with negative energy. Consequently,u0 is also a local minimum of Gλ, and Gλ(u0) < 0.Step 4. If λ < λ0, Gλ has the geometry of the mountain pass lemma.Recalling together Step 1 and Step 2, it can be proved that for λ small enough, Gλsatisfies the geometry of the mountain pass lemma.Consider now u0 the local minimum obtained in Step 3 such that Gλ(u0) < 0 andconsider v ∈ W 2,2
0 (�) with ||�v||L2(�) > rmax and such that Gλ(v) < Gλ(u0). Wedefine
= {γ ∈ C([0, 1],W 2,2
0 (�))
| γ (0) = u0, γ (1) = v},and the mini-max value
c = infγ∈ max
t∈[0,1] Gλ[γ (t)].
Applying the Ekeland variational principle (see [8]), there exists a Palais–Smalesequence {un}n∈N ⊂ W 2,2
0 (�) to the level c.
Step 5. If {un}n∈N ⊂ W 2,20 (�) is a Palais–Smale sequence for Gλ at the level c,
then there exists C > 0 such that ||�un ||L2(�) < C . This can be proved reproducingStep 4 of the proof of Theorem 1.1.Step 6. Since Gλ satisfies the Palais–Smale condition, we can conclude that u =limn→∞ un is a mountain pass type solution to the problem (3.5).
(ii) 1 < p < ∞: Notice that the term corresponding to S2(D2u) plays the role of asuperlinear power in the functional. Hence, the term λ|u|p−1u, that in this case will besuperlinear too, adds nothing new to the problem (1.2) concerning to solvability andmultiplicity. The proof follows identically the outline of the proof of Theorem 1.1, sowe will skip it.
(iii) p = 1: In this case, the associated functional is
Gλ(u) = 1
2
∫
�
|�u|2 −∫
�
∑
1≤i< j≤N
∂i j u∂i u∂ j u − λ
2
∫
�
|u|2,
and the first thing we remark is that Gλ satisfies
Gλ(u) ≥(
1
2− λ
2C1
)
‖�u‖2L2(�)
− C2‖�u‖3L2(�)
,
where C1 is the constant that appears in the Poincaré inequality
⎛
⎝∫
�
|u|2 dx
⎞
⎠
12
≤ C1/21
⎛
⎝∫
�
|�u|2 dx
⎞
⎠
12
.
Hence, if λ < 1C1
, the geometry of the functional is essentially the same as in thesuperlinear case. However, if we remind that the first eigenvalue of the bilaplacianminimizes the Rayleigh quotient, that is,
123
F. Ferrari et al.
λ1 = infu∈W 2,2
0 (�),u �=0
{∫�
|�u|2 dx∫�
|u|2 dx
}
,
then this implies that necessarily λ < λ1.If this condition holds, in a very close way to the previous case, we can prove theexistence of a non trivial solution as a consequence of the mountain pass theorem.
�
4 The homogeneous problem with Navier conditions: bifurcation
In the previous section, since we had Dirichlet boundary conditions, we could work with thevariational formulation of our problems, and hence the existence results were fully based onvariational techniques. However, in this section we deal with Navier boundary conditions,case where we do not have a variational formulation. Indeed, if we consider the problem
{�2u = S2(D2u), in �,
u = �u = 0 on ∂�.
We do not even know whether a nontrivial solution exists. Nonetheless, if we add a reactionterm, i.e., we try to study the problem (1.3), we can use the bifurcation theory to achieveinformation about the solvability of this problem. More precisely, we want to use the Rabi-nowitz Global Bifurcation Theorem. For the sake of completeness, we state this theorem,introducing some necessary notation before.
Let X be a Banach space, A ∈ L(X) and T ∈ C1(X, X), and set
Iλ(u) = u − λAu − T u, u ∈ X. (4.1)
Define� = {(λ, u) ∈ R × X, u �= 0 : Iλ(u) = 0}. Then, the Rabinowitz global bifurcationtheorem (see [3,21]) can be stated as follows.
Theorem 4.1 (Rabinowitz, 1970) Let A ∈ L(X) be compact and let T ∈ C1(X, X) becompact and such that T (0) = 0 and T ′(0) = 0. Suppose that λ∗ is a characteristic valueof A with odd multiplicity. Let C be the connected component of � containing (λ∗, 0). Theneither
(i) C is unbounded in R × X, or(ii) there exists λ̂ ∈ ρ(A)\{λ∗} such that (λ̂, 0) ∈ C.
� In order to apply this theorem to our problem, previously we need to study the eigenvalue
problem associated to �2,
(E P)N
{�2u = λu, in �,
u = �u = 0 on ∂�.
The main particularity of these boundary conditions is that we can reformulate this problemas the following two Dirichlet problems for the laplacian,
(N1)
{−�u = v, in�,
u = 0, on ∂�(N2)
{−�v = λu, in�,
v = 0, on ∂�,
123
Biharmonic elliptic problems
so we can apply all the well known machinery related to this kind of problems. Conse-quently,
Proposition 4.2 Let λ1 be the first eigenvalue of the problem (E P)N . Then, λ1 is simple,i.e., (E P)N has a positive solution and the set of all solutions is a one dimensional linearsubspace of W 2,2(�) ∩ W 1,2
0 (�).
Proof Consider the problem
( ˜E P)
{−�u = μu in�,
u = 0 on ∂�.
It is well known that the first eigenvalue μ1 of this problem is simple, and moreover,that there exists an orthonormal basis in L2(�) composed of eigenfunctions φk ∈ W 1,2
0 (�),k ∈ N, with their corresponding eigenvalues μk .
Since the domain � is supposed to be smooth, this eigenfunctions are actually C∞(�),and then it is easy to check that λk = μ2
k are eigenvalues of (E P)N , and φk the associatedeigenfunctions.
Using the orthogonality and the completeness of the basis {φk} and the fact that μ1 issimple for ( ˜E P), it easily follows that indeed λ1 = μ2
1 is simple too for the problem (E P)N .�
Now we can apply the Rabinowitz Theorem to prove the Theorem 1.2.
Proof of Theorem 1.2 Take X = W 2,2(�)∩ W 1,20 (�), and consider Au = �−2u and T u =
�−2(S2(D2u)). Hence,
�2u = S2(D2u)+ λu (4.2)
is equivalent to
Iλ(u) = 0. (4.3)
To apply the Rabinowitz Theorem we first need to know that A and T are compact fromW 2,2(�) ∩ W 1,2
0 (�) to W 2,2(�) ∩ W 1,20 (�). Consider the problem
{�2u = S2(D2ϕ)+ λϕ =: F(ϕ), in �,
u = �u = 0 on ∂�.
Since we have Navier conditions, we can split it into{
−�u = v, in�,
u = 0, on ∂�
{−�v = F(ϕ), in�,
v = 0, on ∂�.
Suppose ϕ ∈ W 2,2(�)∩W 1,20 (�). Then S2(D2ϕ) ∈ L1(�) and therefore F(ϕ) ∈ L1(�).
Hence, considering the second problem, we deduce v ∈ W 1,q0 (�) with q < 3
2 , and therefore
u ∈ W 3,q(�) ∩ W 1,20 (�). As a consequence D2u ∈ W 1,q(�), and by Rellich–Kondrachov
theorem, this space is compactly embedded in L p(�), for p < 3q3−q . In particular, taking
q so close enough to 3/2, we can choose p = 2, that is, W 1,q(�) ⊂⊂ L2(�), and henceD2u ∈ L2(�), that is, u ∈ W 2,2(�) ∩ W 1,2
0 (�). Thus,
�−2 F : W 2,2(�) ∩ W 1,20 (�) → W 2,2(�) ∩ W 1,2
0 (�) (4.4)
is a compact operator, so A and T are also compact operators.
123
F. Ferrari et al.
On the other hand, we need to prove that T (0) = 0 and T ′(0) = 0. Notice that T (0) =�−2(0). So we get that T (0) = 0 by splitting the problem and applying the maximumprinciple. For the derivative, it can be checked that
dT (u + tϕ)
dt
∣∣∣∣t=0
= �−2
⎛
⎝∑
1≤i< j≤N
∂i i u∂ j jϕ + ∂i iϕ∂ j j u − 2∂i j u∂i jϕ
⎞
⎠ (4.5)
and therefore we conclude T ′(0) = 0.Finally, by Proposition 4.2, we know that the first eigenvalue λ1 associated to A in �
is simple, that is, it has multiplicity one. Thus we can apply Theorem 4.1 to conclude thatthe connected component of � that contains (λ1, 0) is unbounded, what follows from anapplication of Theorem 2.12 in [21]. � Remark 4.3 Notice that in dimension N = 4 we do not have compactness. For N > 4 thesituation is even worse, because the inverse operator �−2 F has a range in general greaterthan W 2,2(�).
Remark 4.4 We cannot use the Rabinowitz Bifurcation Theorem in the case of Dirichletboundary conditions because the first eigenvalue is not simple in general. In fact, examplesof domains with associated eigenvalues of even multiplicity can be found in the literature,see for instance [7]. However, Ortega and Zuazua proved in [20] that there exists a smalldeformation u ∈ W 5,∞(�) of the domain, such that all the eigenvalues in the deformeddomain � + u are simple. That is, in particular and in this sense, the simplicity of the firsteigenvalue with Dirichlet condition is generic.
5 The nonhomogeneous problem with Navier conditions: fixed points arguments
From the point of view of the applications the nonhomogeneous problem is very relevant.Indeed, the aim of this section is to prove Theorem 1.3, that states an existence result forproblem (1.4) for dimension N = 3.
Note that if we denote ∇i j as the two dimensional vector made up of the i , j coordinates ofthe gradient vector, we can reformulate the two dimensional result proved in [12] as follows.
Lemma 5.1 For any functions v1, v2 ∈ W 1,2(�) and v3 ∈ W 1,20 (�)∩W 2,2(�) the following
equality is fulfilled∫
det (∇i jv1,∇i jv2)v3 dx =∫
v1∇i jv2 · ∇⊥i j v3 dx
where ∇⊥i j v3 = (∂ jv3,−∂iv3).
With this technical lemma we will be able to prove Theorem 1.3.
Proof First, consider the linear problems{�2u1 = S2(D2ϕ1)+ μ f (x) in �,u1 = 0, �u1 = 0, on ∂�,
(5.1)
and{�2u2 = S2(D2ϕ2)+ μ f (x) in �,u2 = 0, �u2 = 0, on ∂�,
(5.2)
123
Biharmonic elliptic problems
where ϕ1, ϕ2 ∈ W 1,20 (�) ∩ W 2,2(�). These problems are solvable, and we find u1, u2 ∈
W 1,20 (�) ∩ W 2,2(�).
Notice that if N = 3 and ϕi ∈ W 2,2(�) ∩ W 1,20 (�), i = 1, 2, we have that with the only
information S2(D2ϕi ) ∈ L1(�) is enough to start. Indeed, we have ui ∈ W 3,q(�)∩W 1,20 (�),
with q < 32 and then �ui ∈ L2(�).
Substracting both equations we obtain{�2(u1 − u2) = S2(D2ϕ1)− S2(D2ϕ2) in �,u1 − u2 = 0, �(u1 − u2) = 0, on ∂�.
(5.3)
Again, following [12], we know that
det(D2i j (ϕ1))− det(D2
i j (ϕ2)) = det{∇i j (∂iϕ1), ∇i j [∂ jϕ1 − ∂ jϕ2]}+ det{∇i j [∂iϕ1 − ∂iϕ2], ∇i j (∂ jϕ2)},
and therefore, by Lemma 5.1, if w ∈ W 2,2(�) ∩ W 1,20 (�)
〈S2(D2ϕ1)− S2(D
2ϕ2), w〉 =∑
1≤i< j≤N
〈det(D2i j (ϕ1))− det(D2
i j (ϕ2)), w〉
=∑
1≤i< j≤N
〈det{∇i j (∂iϕ1), ∇i j [∂ jϕ1 − ∂ jϕ2]} + det{∇i j [∂iϕ1 − ∂iϕ2], ∇i j (∂ jϕ2)}, w〉
=∑
1≤i< j≤N
∫
[∂iϕ1∇i j (∂ jϕ1 − ∂ jϕ2)− ∂ jϕ2∇i j (∂iϕ1 − ∂iϕ2)] · ∇⊥i jw dx .
Consequently,
|〈S2(D2ϕ1)− S2(D
2ϕ2), w〉|≤
∑
1≤i< j≤N
∫
[|∇ϕ1||D2i j (ϕ1 − ϕ2)| + |∇ϕ2||D2
i j (ϕ1 − ϕ2)|]|∇i jw| dx
≤∑
1≤i< j≤N
∫
(|∇ϕ1| + |∇ϕ2|)|D2(ϕ1 − ϕ2)||∇w| dx
≤ C(n)(‖∇ϕ1‖L4(�) + ‖∇ϕ2‖L4(�))‖D2(ϕ1 − ϕ2)‖L2(�)‖∇w‖L4(�),
and finally, using the Sobolev embedding for W 1,20 (�),
|〈S2(D2ϕ1)− S2(D
2ϕ2), w〉|≤ C(n)(‖�ϕ1‖L2(�) + ‖�ϕ2‖L2(�))‖�(ϕ1 − ϕ2)‖L2(�)‖�w‖L2(�).
If now we multiply the equation of problem (5.3) by (u1 − u2), and we integrate by partstwice the left hand side, we achieve∫
|�(u1 − u2)|2 dx ≤ |〈S2(D2ϕ1)− S2(D
2ϕ2), u1 − u2〉|≤ C(n)(‖�ϕ1‖L2(�)+‖�ϕ2‖L2(�))‖�(ϕ1−ϕ2)‖L2(�)‖�(u1−u2)‖L2(�),
that is,
‖�(u1 − u2)‖L2(�) ≤ C(‖�ϕ1‖L2(�) + ‖�ϕ2‖L2(�))‖�(ϕ1 − ϕ2)‖L2(�). (5.4)
123
F. Ferrari et al.
Consider now v to be the solution to the problem{�2v = μ f (x) in �,v = 0, �v = 0, on ∂�.
(5.5)
By the Sobolev embedding, there holds
‖�v‖L2(�) ≤ μ‖ f ‖L1(�).
Take ρ > 0, and suppose that ϕ1, ϕ2 ∈ Bρ(v) := {ϕ ∈ W 2,2(�) ∩ W 1,20 (�) : ‖�(v
− ϕ)‖L2(�) ≤ ρ}. Hence, equation (5.4) becomes
‖�(u1 − u2)‖L2(�) ≤ C(ρ + ‖�v‖L2(�))‖�(ϕ1 − ϕ2)‖L2(�)
≤ 12‖�(ϕ1 − ϕ2)‖L2(�)
(5.6)
for μ and ρ small enough. Moreover, for both i = 1 and i = 2,
‖�(ui − v)‖2L2(�)
=∫
�(ui − v)�(ui − v) dx =∫
S2(D2ϕi )(ui − v) dx
≤ ‖ui − v‖L∞(�)‖S2(D2ϕi )‖L1(�). (5.7)
By (5.7) and the Sobolev embedding,
‖�(ui − v)‖2L2(�)
≤ C̃‖�(ui − v)‖L2(�)‖�ϕi‖2L2(�)
.
On the other hand,
‖�ϕi‖2L2(�)
≤ 2(‖�(ϕi − v)‖2
L2(�)+ ‖�v‖2
L2(�)
)
≤ 2(ρ2 + μ2‖ f ‖2
L1(�)
)≤ ρ
C̃
for ρ and μ small enough. Thus,
‖�(ui − v)‖L2(�) ≤ ρ, (5.8)
that is, ui ∈ Bρ(v). Therefore, if we define the operator
T : Bρ(v) → Bρ(v)ϕ → T (ϕ) = u,
where u is the solution to the problem{�2u = S2(D2ϕ)+ μ f (x) in �,u = 0, �u = 0, on ∂�,
(5.9)
inequality (5.6) assures that T is contractive, and hence, by Banach Fixed Point Theorem, ithas a unique fixed point u = T (u), which is a solution to problem (1.4). � Remark 5.2 This argument can be exactly reproduced in the case of Dirichlet boundaryconditions for N = 3. Moreover, thanks to the Sobolev embedding, the same result can beobtain if we add a reaction term |u|p−1u, with p > 1, in problem (1.4).
Remark 5.3 This proof cannot be extended to the case of dimension N = 4, because itstrongly depends on the inclusion W 2,2(�) ⊂ L∞(�). Indeed, for dimension N = 3, theembedding W 2,2(�) ⊂ L p(�) is obtained for all p ≤ +∞. However, the inclusion in theextremal case p = +∞ is not reached for N = 4.
123
Biharmonic elliptic problems
6 Some open problems
We enumerate a collection of open problems emerged from this work that seem to be inter-esting.
• To deal with the lack of compactness at the gradient level in dimension N = 4 when weare working with the variational approach of the problem (1.2) (see Remark 3.2).
• To study the behavior of the unbounded branch of solutions obtained in Theorem 1.2: ifit bifurcates towards λ > λ1 or λ < λ1, if the branch crosses λ = 0 giving a nontrivialsolution for this value of the parameter, etc.
• To search for new fixed point arguments to reproduce the result of section 5 in the case ofdimension N = 4 (see Remark 5.3).
Acknowledgments F. Ferrari and I. Peral wish to thank the Department of Mathematics of the UniversidadAutónoma de Madrid and to the department of Mathematics of the Universitá di Bologna, respectively, for itskind hospitality.
References
1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partialdifferential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727(1959)
2. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partialdifferential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92(1964)
3. Ambrosetti, A., Malchiodi, A.: Nonlinear analysis and semilinear elliptic problems. In: Cambridge Studiesin Advanced Mathematics, vol. 104, pp 60–65 (2006)
4. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J.Funct Anal. 14, 349–381 (1973)
5. Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)6. Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math.
Pures Appl. 72, 247–286 (1993)7. Coffman, C.V., Duffin, R.J., Shaffer, D.H.: Constructive Approaches to Mathematical Models, pp. 267–
277. Academic Press, New York (1979)8. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)9. Ehrenpreis, L.: Solution of some problems of division. I. Division by a polynomial of derivation. Am. J.
Math. 76(4), 883–903 (1954)10. Ehrenpreis, L.: Solution of some problems of division. II. Division by a punctual distribution. Am. J.
Math. 77(2), 286–292 (1955)11. Escudero, C.: Geometric principles of surface growth. Phys. Rev. Lett. 101, 196102 (2008)12. Escudero, C., Peral, I.: Some fourth order nonlinear elliptic problems related to epitaxial growth. J. Differ.
Equ. 254, 2515–2531 (2013)13. Ferrari, F.: Ground state solutions for k−th Hessian operators. Boll. Un. Mat. Ital. B (7) 9, 553–586 (1995)14. Ferrari, F., Franchi, B., Verbitsky, I.E.: Hessian inequalities and the fractional Laplacian. J. Reine Angew.
Math. 667, 133–148 (2012)15. Ferrari, F., Verbitsky, I.E.: Radial fractional Laplace operators and Hessian inequalities. J. Differ. Equ.
253(1), 244–272 (2012)16. García Azorero, J., Peral, I.: Multiplicity of solutions for elliptics problems with critical exponents or
with a non-symmetric term. Trans. Am. Math. Soc. 323(2), 877–895 (1991)17. Gazzola, F., Grunau, H., Sweers, G.: Polyharmonic boundary value problems. Positivity preserving and
nonlinear higher order elliptic equations in bounded domains. In: Lecture Notes in Mathematics, 1991.Springer, Berlin (2010)
18. Malgrange, B.: Existence et approximation des solutions des quations aux drives partielles et des quationsde convolution. Ann. Inst. Fourier 6, 271–355 (1955–56)
19. Marsili, M., Maritan, A., Toigo, F., Banavar, J.R.: Stochastic growth equations and reparametrizationinvariance. Rev. Mod. Phys. 68, 963–983 (1996)
123
F. Ferrari et al.
20. Ortega, J.H., Zuazua, E.: Generic simplicity of the spectrum and stabilization for a plate equation. SiamJ. Control Optim. 39(5), 1585–1614 (2001)
21. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513(1971)
22. Stein, E.M.: Singular integrals and differentiability properties of functions. In: Princeton MathematicalSeries, No. 30. Princeton University Press, Princeton, N.J. (1970)
23. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Prince-ton University Press, Princeton (1993)
24. Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables. I. The theory of H p
-spaces. Acta Math. 103, 25–62 (1960)25. Tartar, L.: Nonlinear Analysis and Mechanics. Heriott-Watt Symposium, IV. Pitman, London, 197926. Tartar, L.: Systems of Nonlinear Partial Differential Equations. Reidel, Dordrecht (1983)27. Tso, K.: Remarks on critical exponents for hessian operators. Annales Inst. H. Poincaré Anal. Non Linéaire
7, 113–122 (1990)
123
