On a fourth-order elliptic equation involving the critical Sobolev exponent: The effect of the graph topology

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Abstract

In this paper, we consider the following nonlinear equation Δ2u=Kun+4n4+εu, with u>0 in Ω and u=Δu=0 on Ω, where Ω is a smooth bounded domain in Rn, n>8, K is a smooth positive function in Ω and ε is a small positive parameter. Using the effect of the graph topology of the function K, we prove existence, multiplicity and concentration of solutions. We prove also a nonexistence result.

Introduction

The main purpose of this paper is to establish the existence of solutions for the following fourth-order equation involving the critical Sobolev exponent under Navier boundary conditions: {Δ2u=Ku21+εu,u>0 in Ω,Δu=u=0on Ω, where ε is a small nonnegative number, Ω is a bounded domain in Rn,n>8, with smooth boundary Ω, K is a smooth positive function in Ω and 2=2nn4 is the critical exponent for the Sobolev embedding H2H01(Ω)L2(Ω).

The interest in this problem grew up from its resemblance to some fourth-order type of equations arising in conformal geometry. A well-known example is the problem of prescribing Paneitz curvature (so-called Q-curvature): given a function K defined in Riemannian manifold (M,g) of dimension n5, we ask whether there exists a metric g̃ conformal to g such that K is the Paneitz curvature of the new metric g̃. We can refer to [1], [2], [3], [4], [5], [6], [7], [8], [9] and the references therein for details.

In contrast to the concentration phenomena for the second order elliptic problems with the critical Sobolev exponent which has been extensively investigated in the recent years, (see for example [10], [11], [12], [13], [14], [15], [16], [17]), fewer results are available to the concentration phenomena for the fourth-order elliptic problems (see for example [18], [19]).

In the special case where K1, a first result concerning (1.1) has been obtained by Van der Vorst [20], who proved the existence of the solution of (1.1) for ε]0,λ12[, where λ1 denotes the first positive eigenvalue of Δ on H2H01(Ω), generalizing to the fourth-order elliptic problem, the famous result of Brezis–Nirenberg [11]. Van der Vorst proved also in [21] a nonexistence result of (1.1) when ε=0 and Ω is a star shaped domain. Such nonexistence result has been proved by Pohozaev [22] for the second order elliptic problems. In the same case, (ε=0 and K1), F. Ebobisse and M. Ould Ahmedou [23] investigated the effect of the topology of the domain Ω on the existence of the solution. They proved that (1.1) has a solution provided that some homology group of Ω is nontrivial. This result generalizes to (1.1) the famous result of Bahri–Coron [24] initially obtained for the second order elliptic problem. In [18], El Mehdi and Selmi have constructed a solution of (1.1) which concentrates around a critical point of Robin’s function as ε0. Such concentration result extends to fourth order equations some results obtained by O. Rey (see [25], [16]) for the second order elliptic equations.

However, as far as the author knows, the case where K1 has not been considered before.

Pushing further the resemblance of the two problems, the second order elliptic type problems from one part and the fourth order elliptic type problems from another part, we investigate the effect of the graph topology of the function K on the existence of a solution for (1.1), extending the results obtained by J. Chabrowski and S. Yan [26], to the case of the fourth-order elliptic problem. Namely, in the first part of this paper, we will present the effect of the topological structure of the graph of the function K on the existence of the solution of (1.1) as ε0. In the second part, we construct solutions of (1.1) which blow up and concentrate around a boundary or interior local maximum point of K as ε tends to zero. Then, we estimate the number of such solutions using the Lusternik–Schnirelmann category of the set of local maxima of K. In the last part, we will prove the nonexistence of solutions which concentrate around a non degenerate critical point of K. Compared with the second order case, further technical problems arise. In fact, in our case we cannot use, as in the Laplacian case, the method of moving planes in order to show that blow up points are away from the boundary of the domain. This is due to the fact that the Navier boundary condition is not invariant under the Kelvin transformation of a biharmonic operator. To overcome such a difficulty, we perform refined expansions of the Euler functional associated to (1.1) and its gradient near a neighborhood of highly concentrated functions. Such expansions, which are of independent interest, are highly nontrivial and use the techniques developed by A. Bahri [27] and O. Rey [16] in the framework of the “Theory of critical points at infinity”.

To state our results, we need to introduce the following notations. Let M={xΩ;K(x)=Kmax=maxxΩK(x)}. Using the topological structure of the global maximum set of K, we have the following first result.

Theorem 1.1

For n>8, assume the following.

  • (1)

    M is not contractible in a small neighborhood of itself, but M is contractible in {x;K(x)t} for some constant t belonging to(Kmax24/(n4),Kmax),such that maxxΩK(x)<t.

  • (2)

    For each critical point x of K such that Kmax>K(x)t, we haveΔK(x)>0.

Then, for each ε[0,ε0],(1.1) has a solution uε converging (up to subsequence) strongly in H2H01(Ω) as ε0.

The proof of Theorem 1.1 goes along the idea of [28] to construct a solution whose energy is strictly greater than Sn/4/nKmax(n4)/4, where S is the best Sobolev constant (see (1.3)). Above this energy level, the corresponding functional does not satisfy the Palais–Smale condition. To overcome this difficulty, we first perturb the original problem suitably and construct a solution for this perturbed problem, whose energy is strictly greater than Sn/4/nKmax(n4)/4. Then we prove that the solution for the perturbed problem (see (3.1) below) converges strongly in H2H01(Ω) to a solution of the original problem (1.1).

Unlike Theorem 1.1, where the set of global maxima of K was used, the next two results use only local information. Namely, we will give the concentration and multiplicity results for problem (1.1) using the topological structure of the local maximum set of the function K. For this purpose, we recall that the Lusternik–Schnirelmann category of the subset FΩ denoted by catΩ(F) is the least integer k such that F can be covered by k closed subsets A1,,Ak of F such that for all i,Ai is contractible in Ω. If no such integer exists, then by definition catΩ(F)=. If F=Ω we write catF(F)=cat(F). We need also to fix the following notations. For λ>0 and xRn, let δx,λ(y)=cnλn42(1+λ2|yx|2)n42,cn=[(n4)(n2)n(n+2)](n4)/8. It is well known (see [29]) that δx,λ are the only solutions of Δ2u=un+4n4,u>0 in Rn with uL2(Rn) and ΔuL2(Rn). They are also the only minimizers of the Sobolev inequality on the whole space; that is, S=inf{ΔuL2(Rn)2uL2nn4(Rn)2:ΔuL2,uL2nn4,u0}. We denote by Pδx,λ the projection of the δx,λ’s onto H2(Ω)H01(Ω), defined by Δ2Pδx,λ=Δ2δx,λin ΩandΔPδx,λ=Pδx,λ=0on Ω, and we set φx,λ=δx,λPδx,λ. We denote by G Green’s function of Δ2, that is, for all xΩ, Δ2G(x,)=cnδx̃in ΩΔG(x,)=G(x,)=0on Ω, where δx̃ denotes the Dirac mass at x and cn=(n4)(n2)|Sn1|. We also denote by H the regular part of G, that is, H(x,y)=|xy|4nG(x,y),for (x,y)Ω×Ω. The space H(Ω)H2(Ω)H01(Ω) is equipped with the norm and its corresponding inner product , defined by u=(Ω|Δu|2)1/2,uH(Ω),u,v=ΩΔuΔv,u,vH(Ω). For xΩ, λ>0, let Ex,λ={vH(Ω):v,Pδx,λ=v,Pδx,λλ=v,Pδx,λxj=0,j=1,,n}, where the xj is the j-th component of x.

In the following two results, we construct a solution for (1.1) which concentrates at a local maxima of K as ε0.

Theorem 1.2

For n>8, let M be a connected closed set in Ω satisfyingK(x)KMR,xM.Let K be a positive function satisfying:

  • (1)

    K(x)KMa(d(x,M))kd(x,M)δ,

  • (2)

    |DiK(x)|=O(d(x,M)k)d(x,M)δ,i=2,[k],

where a and k>4+16n8 are some positive constants. Then, for each ε[0,ε0], problem (1.1) has at least Cat(M) solutions uε of the formuε=αεPδxε,λε+vε,where vεExε,λε and as ε0, we haveαεKM1/(22),vε0,xεx0M,λε,λεd(xε,Ω).

Theorem 1.3

For n>8, let M be a connected closed set in Ω satisfyingK(x)KMR,xM.Let M be a connected closed set compactly contained in Ω such that MM and satisfying

  • (1)

    KM>maxxMK(x),

  • (2)

    M{x;K(x)=KM}=M,

  • (3)

    M{x;K(x)>KM}=,

  • (4)

    |DiK(x)|=O(εk), for each i=2,3,4 and for all xM,

where k>6 is some positive constant. Then for each ε[0,ε0], problem (1.1) has at least CatM(M) solutions uε of the formuε=αεPδxε,λε+vε,where vεExε,λε and as ε0, we haveαεKM1/(22),vε0,xεx0M,λε.

Remark 1.4

Similar conditions to those given in Theorem 1.2, Theorem 1.3, have been imposed on a global maximum point in [30], in order to obtain the existence of one solution in the case of the Laplacian problem.

In contrast with the previous results where we gave existence and multiplicity results, we give next a nonexistence result. Namely, we show that this problem has no solutions that concentrate around a non degenerate critical point of K as ε approaches 0.

Theorem 1.5

For n>8, let x0Ω be a critical point of K such that ΔK(x0)0 . Then, there is no solution of (1.1) of the form (1.6) satisfying (1.7), (1.8), (1.9), (1.10).

Remark 1.6

We believe our results, (Theorem 1.1, Theorem 1.3, Theorem 1.5), to be true for n=8. More careful estimates similar to those developed by Rey [31] in the case of Laplacian, should include the case n=8. However, in Theorem 1.2 we should assume that n>8 in order to make the conditions of this result meaningful. The assumption n8 is sharp, since for 5n7, (1.1) has no solution when Ω is a ball for ε small [32].

We organize the remainder of the present paper as follows. In Section 2 we set up the variational structure and give some careful expansion of the Euler functional associated to problem (1.1). Sections 3 Proof of, 4 Proof of will be devoted to the proof of the existence results. The nonexistence result will be proved in Section 5. To prove our results, we need some technical lemmas whose proofs are collected in the Appendix.

Section snippets

Variational structure and technical framework

Problem (1.1) enjoys a variational structure. Indeed, for ε>0, solutions of (1.1) correspond to positive critical points of the functional Jε defined on H(Ω) by: Jε(u)=Ω|Δu|2εΩu2(ΩK(y)|u|2nn4)n4n. We define the following subset of H(Ω)Mε={(x,λ,v)Ω×R+×H(Ω):λ>1ν0,vEx,λ}, where ν0 is a small positive constant. Let us define the functional ψε:MεR,ψε(x,λ,v)=Jε(Pδx,λ+v). It is well known that (x,λ,v)Mε is a critical point of ψε if and only if u=Pδx,λ+v is a critical point of Jε, (see for

Proof of Theorem 1.1

Let τ0 be a small constant. For each fixed small ε>0, we consider the following perturbed problem {Δ2u=K(x)u21τ+εuu>0 in Ω,u=Δu=0on Ω. The solutions of (3.1) are critical points on H(Ω) of the functional Iτ(u)=12Ω(|Δu|2εu2)12τΩK(x)|u|2τ.

It is easy to check that the energy of the solution of (3.1) in the case ε=0 is at least Sn/4/nKmax(n4)/4. Following the idea of [28], to construct a solution for (3.1) whose energy is strictly greater than Sn/4/nKmax(n4)/4, we have the following

Proof of Theorem 1.2

First, we define λε,x=[H(x,x)ε]1/(n8). Let =1/(k4)+τ with τ>0 small enough. It is easy to see that (k4)>1,(k4(n4)n8)<1+4n8. Let Nε={x:d(x,M)ε}{x;d(x,Ω)εL},Dε={(x,λ):xNε,λ[ηλε,x,εT]}, where η is a small fixed constant, L> and T are large constants. Without loss of generality, we may assume KM=1. Let cε=Sn12/2(1τε1+4[1+t(n4)]/(n8)), where t<L is a large constant to be determined later. For (x,λ)Dε, we denote F(x,λ)=ψε(x,λ,vε(x,λ))=Jε(Pδx,λ+vε(x,λ)). Using Proposition 2.1,

Proof of Theorem 1.5

Arguing by contradiction, we suppose that (1.1) has a solution of the form uε=αεPδxε,λε+vε satisfying (1.7), (1.8), (1.9), (1.10). First, multiplying (1.1) by vε and integrating over Ω, we get Ω|Δvε|2=ΩK(y)|αεPδxε,λε+vε|21vε+εΩ(αεPδxε,λε+vε)vε=αε21ΩK(y)Pδxε,λε21vε+(21)αε22ΩK(y)Pδxε,λε22vε2+εΩ(αεPδxε,λε+vε)vε+O(vε2+σ), where σ>0 is a constant. Observe that, ΩK(y)Pδxε,λε22vε2=ΩK(y)(δxε,λε22+O(δxε,λε23φxε,λε+φxε,λε22))vε2. Using Lemma A.1, it is easy to check that Ωδx

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