Existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions

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Abstract

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.

Introduction

In this paper, we are concerned with the following quasilinear elliptic equations with a mixed nonlinear boundary condition {div(a(x)|u|p2u)+|u|q2u=λg(x)|u|r2u,in Ω,a(x)|u|p2νu+b(x)|u|p2u=0,on Ω. It is assumed that Ω is a smooth exterior domain in RN(N3), that is, Ω is the complement of a bounded domain with C1,δ(0<δ<1) boundary Ω,ν is the unit outer normal to Ω,1<p<N, and λ>0 is a real parameter.

In recent years, existence, non-existence and multiplicity of solutions for problems involving the p-Laplace operator were studied by several authors, in bounded domain, see e.g. [1], [2], [3], [4], [5], while in unbounded domains or the entire RN, see e.g. [6], [7], [8], [9], [10], [11] and the references therein. Problems of this type appear in many nonlinear diffusion problems, such as the mathematical model of non-Newtonian fluid [12], the chemotactic aggregation model introduced by Keller and Segel [13]. As pointed out in [14], [15], certain stationary waves in nonlinear Klein–Gordon or Schrodinger equations can be reduced to this form. For some physical motivation of such nonlinear boundary conditions, see [16]. From the point of view of mathematics, similar problems with nonlinear boundary condition appear in the study of the optimal constants for the Sobolev trace embeddings. See e.g. [1], [2], [8], [5].

In this paper, the weight functions a,b and g satisfy the following assumptions:

  • (a)

    0<a0a(x)L(Ω)C0,δ(Ω¯);

  • (b)

    0<b(x)C(Ω);

  • (g)

    0g(x)L(Ω)Lp0(Ω) with p0=p/(pr) where 1<r<p,p=NpNp, and g is positive on a nonempty open subset of Ω.

It is clear that problem (P) possesses a trivial solution u=0 for any λR. We are interested in the existence and multiplicity of weak solutions of (P).

By a weak nontrivial solution of (P), one means a nontrivial function uXELq(Ω) verifying for all ϕX the identity Ωa(x)|u|p2uϕdx+Ωb(x)|u|p2uϕdσ+Ω|u|q2uϕdx=λΩg(x)|u|r2uϕdx where E is the completion of the restriction on Ω of functions of C0(RN) with respect to the norm ua,b=(Ωa(x)|u|pdx+Ωb(x)|u|pdσ)1p, and X is a reflexive Banach space endowed with the norm u=(ua,bp+uLq(Ω)p)1p. Set Lr(Ω;g){u:ΩRu is Lebesgue measurable,Ωg(x)|u|rdx<}. Under the assumptions (a),(b) and (g), we have the compact embedding XLr(Ω;g) for r(1,p), which will be stated in Section 2, so all integrals in (1.1) are well-defined and converge. Define the functional Jλ:XR by Jλ(u)=1pΩa(x)|u|pdx+1pΩb(x)|u|pdσ+1qΩ|u|qdxλrΩg(x)|u|rdx. Then JλC1(X,R) with its derivative Jλ(u),ϕ=Ωa(x)|u|p2uϕdx+Ωb(x)|u|p2uϕdσ+Ω|u|q2uϕdxλΩg(x)|u|r2uϕdx,for all ϕX. Therefore the critical points of Jλ correspond to the weak solutions of (P).

The main feature of problem (P) is that the quasilinear differential operator is affected by a different perturbation which behaves like |u|q2u, where q(1,p). The relation of the power r with respect to the powers p and q affects the solvability of (P). In fact, this relation produces different competition between the different growths of the nonlinearities, which plays a crucial role in the study of geometrical feature of energy functional. In [17], the author studied (P) with Dirichlet boundary condition for q=p and proved that (P) has a weak positive solution. Recently, in an interesting paper [18], under the same assumptions on a,b and g, the cases p<r<q<p and p<q<r<p with suitable parameter λ were treated and the existence of a weak solution for (P) was obtained via minimizing methods and mountain pass arguments. Motivated by the papers [18], [17], we look for the existence of ground state and multiple solutions of (P) and related problem with more general nonlinearity via minimax arguments. We note that a ground state is a nontrivial solution which has the least energy among all the nontrivial solutions.

The purpose of this paper is two-fold. One purpose is to find multiple weak solutions to (P). Another purpose is to consider the critical case of (P), that is, the right side nonlinearity grows with critical exponent. In this case, the compactness of the embedding XLp(Ω) fails, and it seems to be useless to impose hypothesis of (g) type for the critical term. To recover some sort of compactness, in spirit of [19], we consider a perturbation of the critical power. Namely, we study the problem {div(a(x)|u|p2u)+|u|q2u=|u|p2u+λg(x)|u|r2u,in Ω,a(x)|u|p2νu+b(x)|u|p2u=0,on Ω under the same assumptions on weight functions.

Now we state the main theorems in this paper. As to problem (P), we have the following results.

Theorem 1.1

Let 1<q<p<r<p . Then for each λ>0, problem (P) has a nonnegative ground state solution and infinitely many solutions in X.

Theorem 1.2

Let pq<r<p . For each λ>0, (P) has a positive ground state solution and a sequence of solutions uk in X with Jλ(uk) as k.

Theorem 1.3

  • (i)

    Let 1<r<p and r<q<p . For each λ>0,(P) has a nonnegative ground state and a sequence of solutions uk in X with Jλ(uk)<0 and Jλ(uk)0k.

  • (ii)

    Let 1<r<p and 1<qr . There exists a constant λ>0, depending on p,q,r,N,Ω,g, such that for each λ>λ,(P) has a nonnegative ground state solution in X.

As to problem (P), we have the following results.

Theorem 1.4

Let p<r<p and 1<q<r . There exists a constant λ0>0 depending on p,q,r,N,Ω,g, such that if λ>λ0, (Phas at least a nontrivial solution in X.

Theorem 1.5

Let 1<r<p and 1<q<p . There exists a constant λ>0 depending on p,q,r,N,Ω,g, such that for any 0<λ<λ, (Phas a sequence of solutions uk in X with negative energy decreasing to zero as k.

Remark 1.6

  • (i)

    In the settings of Theorem 1.1, Theorem 1.2, Theorem 1.3, it is easy to see that (P) has no nontrivial solution for any λ0.

  • (ii)

    By the regularity results of [18], if p<q, it follows from the strong maximum principle in Section 4.8 of [20] that any nonnegative solution is positive.

  • (iii)

    In the cases of Theorem 1.1, Theorem 1.2, Jλ has mountain pass geometry. As a result, we get infinitely many solutions by symmetry mountain pass theorem [21] or the fountain theorem [22]. In the cases of Theorem 1.3, since the right side term is sublinear, we can show that Jλ is bounded from below and coercive, which is similar to the case p<r<q<p discussed in [18]. By investigating the structure near zero of the functional, we obtain the multiplicity result. It is interesting that if the existence of infinitely many solutions remains true in the case of 1<r<p,1<qr or the case of p<r<q<p.

Our proofs are based on variational methods. There are two main difficulties. On the one hand, the lack of compactness of the Sobolev embeddings on unbounded domains renders variational techniques more delicate. To overcome this difficulty, some of the papers use special function space, such as the radially symmetric function space, which possesses compact embedding; see e.g. [10], [15]. In this paper, the integrality of g and the main assumption 1<r<p ensure that the function space X is compactly embedded in the weighted Lebesgue space Lr(Ω;g). In critical case, we use the concentration compactness method of [23], [24] and follow ideas from [2], [19], [5]. On the other hand, when 1<q<p<p, it seems difficult to get the usual compactness condition, so called (PS) condition. We overcome this difficulty by verifying Cerami condition.

Throughout the paper, by C, we mean a constant that may vary from line to line but remains independent of the relevant quantities. We denote by “ ⇀” weak convergence and by “ →” strong convergence.

The paper is organized as follows. In Section 2, we give preliminary lemmas concerning the embedding results, the compactness conditions and some abstract critical point theorems. In Section 3, we consider the subcritical case and give the proofs of Theorem 1.1, Theorem 1.2, Theorem 1.3. We consider the critical case and give the proofs of Theorem 1.4, Theorem 1.5 in the last section.

Section snippets

Preliminary results

We begin with the following result about space embeddings, which are stated in [18] for r(p,p). Here we need to extend this result to r(1,p) and give the details of the proof for the reader’s convenience.

Lemma 2.1

Assume that (a),(b) and (g) hold and q,r(1,p) . Then

  • (i)

    the embedding ELp(Ω) is continuous,

  • (ii)

    the embedding ELr(Ω;g) is continuous, in particular there is C1>0 such thatuLr(Ω;g)C1ua,b,

  • (iii)

    the embedding XLs(Ω)(qsp) is continuous,

  • (iv)

    the embedding XLr(Ω;g) is compact.

Proof

Let D1,p(Ω) be the

The subcritical case

In this section, we consider the subcritical cases and give the proofs of Theorem 1.1, Theorem 1.2, Theorem 1.3. We have the following compactness conditions.

Lemma 3.1

Assume that (a),(b),(g) hold and λ>0.

  • (i)

    If 1<q<p<r<p, then Jλ satisfies (C) condition in X.

  • (ii)

    If p<q<r<p, then Jλ satisfies (PS) condition in X.

Proof

(i) Let {un}X be a Cerami sequence such that for some C>0|Jλ(un)|C,(1+un)Jλ(un)0,as n. We first show that {un}X is bounded in X. In fact, by q<p<r, we have C+o(1)Jλ(un)1rJλ(un),un=(1p1r)

The critical case

In this section, we deal with critical case, that is, problem (P), and give the proof of  Theorem 1.4, Theorem 1.5. Therefore, the functional Jλ is written as Jλ(u)=1pua,bp+1qΩ|u|qdx1pΩ|u|pdxλrΩg(x)|u|rdx.

Let D1,p(Ω) be the Sobolev space introduced in Section 2. By (a), we can use the equivalent norm ua(Ωa(x)|u|pdx)1/p. We need the following concentration compactness lemma which is mainly due to Lions [23], [24]; see also [27]. The following version for p2 can be proved by using

Acknowledgments

The first author was supported by NSFC11001008. The third author was supported by KZ201010028027, PHR201106118 and NSFC11171204.

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