AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 834918 10.1155/2013/834918 834918 Research Article Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions http://orcid.org/0000-0002-0515-5410 Huang Jincheng 1, 2 Rossi Julio 1 Department of Mathematics and Physics Hohai University, Changzhou Campus Changzhou 213022 China hhu.edu.cn 2 College of Sciences Hohai University Nanjing 210098 China hhu.edu.cn 2013 24 11 2013 2013 21 05 2013 27 10 2013 27 10 2013 2013 Copyright © 2013 Jincheng Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Existence and multiplicity results are established for quasilinear elliptic problems with nonlinear boundary conditions in an exterior domain. The proofs combine variational methods with a fibering map, due to the competition between the different growths of the nonlinearity and nonlinear boundary term.

1. Introduction

Consider the following quasilinear elliptic problem: (1)-div(a(x)|u|p-2u)+b(x)|u|p-2u=h(x,u),in  Ω,a(x)|u|p-2un+g(x)|u|p-2u=0,on  Γ=Ω, where Ω is a smooth exterior domain in N, 1<p<N, and   n is the unit vector of the outward normal on the boundary Γ=Ω.

Equations of the type (1) arise in many and diverse contexts like differential geometry , nonlinear elasticity , non-Newtonian fluid mechanics , glaciology , and mathematical biology . As a result, questions concerning the solvability of problem (1) have received great attention; see .

For h(x,u)=c(x)|u|q-2u-d(x)|u|s-2u-e(x)|u|t-2u with q,s(1,p*), t>p*, by using the fibering method, Kandilakis and Lyberopoulos  studied the existence of nonnegative solutions for problem (1) in unbounded domains with a noncompact boundary. When h(x,u)=c(x)|u|q-2u-d(x)|u|s-2u with q,s(1,p*), Lyberopoulos  studied the existence versus absence of nontrivial weak solutions for problem (1). Similar consideration can be found in Kandilakis and Magiropoulos . In , Filippucci et al. established existence and nonexistence results for problem (1) via variational methods combined with the geometrical feature, where h(x,u)=λg(x)|u|r-2u-|u|q-2u. Recently, Chen et al.  considered the existence and multiple of solutions for problem (1) by the variational principle and the mountain pass lemma.

Motivated by these findings, we consider the following quasilinear elliptic problem: (2)-div(a(x)|u|p-2u)+b(x)|u|p-2u=f(x)|u|q-2u,in  Ω,a(x)|u|p-2un+g(x)|u|r-2u=0,on  Γ=Ω, where Ω is a smooth exterior domain in N and  n is the unit vector of the outward normal on the boundary Γ=Ω. Since rp, problem (2) is essentially different from problem (1). Using the Nehari manifold and fibering map, Wu  considered problem (2) for p=2; Afrouzi and Rasouli  considered problem (2) for 1<p<N.

Throughout this paper, we make the following assumptions.

1<p<N, 1<q<p*, and 1<r<p*, where p*=Np/(N-p) and   p*=(N-1)p/(N-p).

The function a(x)a0>0 and a(x)L(Ω).

The function b(x)b0>0 and b(x)L(Ω).

The function f(x) satisfies Ωf+:={xΩ:f(x)>0} and f(x)Lδ(Ω) with p/(p-q)δ>q0=p*/(p*-q).

The function g(x)0,  g(x)0 in Γ and g(x)Lσ(Γ) with p/(p-r)σ>r0=p*/(p*-r).

The purpose of this paper is to find existence and multiplicity of nonnegative solutions to problem (2). Our proofs are based on the variational method. The main difficulty is the lack of compactness of the Sobolev embeddings in unbounded domains. To overcome this difficulty, we impose the integrality conditions (H4)-(H5) on f and g to establish compact Sobolev embedding theorems (see Lemmas 3 and 4).

The rest of the paper is organized as follows. In Section 2, we set up the variational framework of the problem and give some preliminaries. Section 3 is devoted to the existence results for problem (2). The multiplicity of nonnegative solutions for problem (2) is considered in the last section.

2. Variational Framework and Some Preliminaries

In this section, we set up the variational framework and give some preliminaries.

Define the weighted Sobolev space E as the completion of C0(N) under the norm (3)uEp=Ω(|a(x)u|p+b(x)|u|p)dx, which is equivalent to the standard one under assumptions (H2)-(H3). Moreover, denote by Lq(Ω;|f|) and Lr(Γ;g) the weighted Lebesgue spaces equipped with the norm: (4)uLq(Ω;|f|)q=Ω|f(x)||u|qdx,uLr(Γ;g)r=Γg(x)|u|rdS, respectively. The definition of the weak solution of problem (2) reads as follows.

Definition 1.

One says uE is a weak solution of problem (2) if (5)Ωa(x)|u|p-2uvdx+Ωb(x)|u|p-2uvdx+Γg(x)|u|r-2uvdS=Ωf(x)|u|q-2uvdx holds for all vE.

The energy functional corresponding to problem (2) is (6)J(u)=1puEp+1rG(u)-1qF(u), where G(u)=Γg(x)|u|rdS, F(u)=Ωf(x)|u|qdx. It is well known that the weak solutions of (2) are the critical points of the energy functional J(·). If u is a critical point of J(·), then necessarily u belongs to the Nehari manifold: (7)𝒩={uE{0}:J(u),u=0}, where (8)J(u),u=uEp+G(u)-F(u). For all u𝒩, we have uF+, where (9)F+={F(u)>0}.

Moreover (10)J(u)=(1p-1q)uEp+(1r-1q)G(u)=(1p-1r)uEp+(1r-1q)F(u)=(1p-1q)F(u)+(1r-1p)G(u).

The variational framework that we adopt is based on the so-called one-dimensional fibering method proposed by Pohozaev . The central idea of this strategy consists in embedding the original variational problem into the “wider” space E1:=×E and then investigating the conditional solvability of the new problem in E1 under an appropriately imposed constraint. To this end, we define the extended functional ϕ:[0,)×E by setting for any vE(11)ϕ(t,v)=J(tv)=tppvEp+trrG(v)-tqqF(v),t0. If u=tv is a critical point of J(·), then necessarily ϕt(t,v)=0; that is, (12)tp-1vEp+tr-1G(v)-tq-1F(v)=0. In particular, if t0, then (12) is equivalent to (13)ψ(t,v)=vEp, where (14)ψ(t,v)=tq-pF(v)-tr-pG(v).

Now, suppose that t=t(v)0 solves (13) for all vE{0}; then tC1(E). Furthermore, if t(v)>0 exists and is unique for all vE{0}, then (13) generates a bijection between E{0} and 𝒩. Moreover, the following proposition holds.

Lemma 2 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

If v is a conditional critical point of Φ(·), under the constraint vE=1, then u=t(v)v is a critical point of J(·), where Φ(v)=J(t(v)v) and t(v) is a nonnegative solution of (13).

In view of Lemma 2, the problem of finding solutions of (2) will be reduced to that of locating the critical point of Φ(·) under the constraint vE=1.

The following compact embedding theorems play an important role in the proof of our main results.

Lemma 3.

Assume (H2)–(H4). Then the embedding ELq(Ω;|f|) is compact.

Proof.

Let uE. Since p/(p-q)δ>q0, it follows that pqδ=qδ/(δ-1)<p*. Let W1,p(Ω) be the standard Banach space endowed with the norm uW1,p(Ω)p=Ω(|u|p+|u|p)dx. By assumptions (H2)-(H3), E~W1,p(Ω). Similar to the proof of [10, Lemma 2] (see also the proof of [14, Theorem 7.9]), we can prove that W1,p(Ω)Lqδ(Ω) is compact and so is ELqδ(Ω). Let S1 be the best trace embedding constant; that is, (15)S1=infuE,u0uEpuLqδ(Ω)p. By Hölder's inequality, we have (16)Ω|f(x)||u|qdx(Ω|f|δdx)1/δ(Ω|u|qδdx)1/δS1-q/p|f|Lδ(Ω)uEq. This shows that the embedding ELq(Ω;|f|) is continuous.

Assume {un} is a bounded sequence in E. Then by the compact embedding ELqδ(Ω), there exist uE and a subsequence of {un} (not relabelled) such that unu strongly in Lqδ(Ω).

By Hölder's inequality again, we infer (17)Ω|f(x)||un-u|qdx(Ω|f|δdx)1/δ(Ω|un-u|qδdx)1/δ0(Ω|f|δdx)1δ(Ω|f|δdx)1δ1as  n. This completes the proof.

Lemma 4.

Assume (H2)-(H3) and (H5). Then the embedding ELr(Γ;g) is compact.

Proof.

Let uE. Since p/(p-r)σ>r0, it follows that prσ=rσ/(σ-1)<p*. Hence the embedding ELrσ(Γ) is compact (see [15, 16]). Let S2 be the best trace embedding constant; that is, (18)S2=infuE,u0uEpuLrσ(Γ)p. By Hölder's inequality, we have (19)Γg(x)|u|rdS(ΓgσdS)1/σ(Γ|u|rσdS)1/σS2-r/pgLσ(Γ)uEr. This shows that the embedding ELr(Γ;g) is continuous.

Assume {un} is a bounded sequence in E. Then by the compact embedding ELrσ(Γ), there exist uE and a subsequence of {un} (not relabelled) such that unu strongly in Lrσ(Γ).

By Hölder's inequality again, we infer (20)Γg(x)|un-u|rdS(ΓgσdS)1/σ(Γ|un-u|rσdS)1/σ0as  n. This completes the proof.

We also need the following mountain pass lemma (see [17, 18]).

Lemma 5.

Let E be a real Banach space and JC1(E,) with J(0)=0. Suppose

there are ρ,α>0 such that J(u)α for uE=ρ;

there is eE,eE>ρ such that J(e)<0.

Define (21)Γ0={γC1([0,1],E)γ(0)=0,γ(1)=e}. Then (22)c=infγΓ0max0t1J(γ(t))α is finite and J(·) possess a (PS)c sequence at level c. Furthermore, if J satisfies the (PS) condition, then c is a critical value of J.

To get multiplicity results, we need the following fountain theorem due to Bartsch  and a critical point theorem in [20, 21].

Let X be a reflexive and separable Banach space. It is well known that there exist ejX and ej*X* (j=1,2,) such that

ei,ej*=δij, where δij=1 for i=j and δij=0 for ij,

X=span{e1,e2,}¯ and X*=span{e1*,e2*,}¯.

For convenience, we write (23)Xj=span{ej},Yk=j=1kXj,Zk=j=kXj¯,j,k=1,2,.

Lemma 6 (fountain theorem [<xref ref-type="bibr" rid="B19">19</xref>]).

Assume JC1(X,1) is an even functional that satisfies the (PS)c condition. If for every k there exist ρk>rk>0 such that

ak:=infuZk,uX=rkJ(u) as k,

bk:=maxuYk,uX=ρkJ(u)0,

then J has a sequence of critical points {uk} with J(uk).

Lemma 7 (see [<xref ref-type="bibr" rid="B20">20</xref>, <xref ref-type="bibr" rid="B21">21</xref>]).

Let JC1(X,1), where X is a Banach space. Assume that J satisfies the (PS)c condition and is even and bounded from below, and J(0)=0. If for any k there exists a k-dimensional subspace Yk and ρk>0 such that (24)supuYk,uX=ρkJ(u)<0, then J has a sequence of critical values ck<0 satisfying ck0 as k.

3. Existence of Nonnegative Solutions

In this section, the existence results are established for problem (2). The proofs combine variational methods with a fibering map. Since J(u)=J(|u|), we may suppose that the solution to problem (2) is nonnegative throughout this paper.

Theorem 8.

Let (H1)–(H5) hold with either q<min{p,r} or q>max{p,r}. Then problem (2) admits a nonnegative nontrivial weak solution uE{0} which is also a ground state.

Proof.

Suppose q<min{p,r}. Rewriting (13) as (25)tp-qvEp+tr-qG(v)=F(v), we immediately see that for every vF+ (where F+ is defined by (9)) there exists a unique t(v)>0 satisfying (25). Moreover, it can be easily checked that (26)μt(μv)=t(v),for  μ>0,vF+,(27)Φ(v)=(1p-1q)(t(v))pvEp+(1r-1q)(t(v))rG(v)<0vF+.

Consider now the variational problem (28)M:=infvF+,vE=1Φ(v)<0. Let {vn} be a minimizing sequence in F+ with vnE=1. Then there exists v0E such that vnv0 in E. By Lemmas 3 and 4, we have F(vn)F(v0)0 and G(vn)G(v0)0.

We first assert that v0F+. Suppose the contrary; then F(v0)=0. In view of (25), (29)F(vn)(t(vn))p-q.

Letting n, it follows that t(vn)0. Thus (30)Φ(vn)=(1p-1q)(t(vn))pvnEp+(1r-1q)(t(vn))rG(vn)0,asn, which contradicts M<0.

Next, we prove v0E=1. If not, then v0E<liminfnvnE=1. So, there exists μ>1 such that μv0E=1. From (25), we have (31)(t(μv0))p-q+(t(μv0))r-qG(μv0)=F(μv0). This and (26) yield (32)μ-p(t(v0))p-q+(t(v0))r-qG(v0)=F(v0).

On the other hand, it follows from (29) that {t(vn)} is bounded and so there exists a subsequence (not relabelled) such that t(vn)t0>0. Thus by (25), we have (33)t0p-q+t0r-qG(v0)=F(v0).

Hence t0<t(v0). Notice that (34)Ψ(t)=(1p-1q)tpv0Ep+(1r-1q)trG(v0) is strictly decreasing for all t>0; we have (35)M=liminfnΦ(vn)Ψ(t0)>Ψ(t(v0))=Φ(v0), which is a contradiction. So, v0E=1 and v0 is a critical point of Φ(·). By Lemma 2, u=t(v0)v0 is a nontrivial solution of problem (2). Since t=t(v) is a unique solution of (25), then (25) generates a bijection between E{0} and 𝒩 and so the obtained solution is actually a ground state.

The case q>max{p,r} can be treated in a similar way.

Remark 9.

Afrouzi and Rasouli  consider the following problem: (36)-div(|u|p-2u)+b(x)|u|p-2u=λf(x)|u|q-2u,in  Ω,|u|p-2un=g(x)|u|r-2u,on  Γ=Ω, where Ω is a bounded domain in N and 1<q<p<r<p*=(N-1)p/(N-p). The functions f and g are continuous functions which change sign in Ω¯. Using the Nehari manifold and fibering map, they proved that problem (36) has at least two nontrivial nonnegative solutions if |λ| is sufficiently small. In fact, by slight modification, we can prove that the result they established is still true if Ω is a smooth exterior domain or the parameters satisfy 1<r<p<q<p*=Np/(N-p). But for g(x)0, we can prove that problem (36) has at least one nontrivial nonnegative solution for |λ| is sufficiently small via the method used in . Notice that our result (Theorem 8) does not need |λ| to be small.

Theorem 10.

Let (H1)–(H5) hold with r<q<p. Then problem (2) has a nonnegative nontrivial weak solution.

Proof.

From Lemma 3, we have (37)F(u)S1-q/pfLδ(Ω)uEq. Thus (38)J(u)1puEp-1qS1-q/pfLδ(Ω)uEq. So J(u) is coercive by p>q.

By Lemmas 3 and 4, it is easy to verify that J is weakly lower semicontinuous. So J has a minimum point u in E and u is a weak solution of (2).

In the following, we prove infuEJ(u)<0. Let vW01,p(Ω)F+. Then (39)J(tv)=1ptpvEp-1qtqF(v). Thus J(tv)<0 for t>0 is sufficiently small. Notice that W01,p(Ω)F+E; we obtain (40)infvEJ(v)infvW01,p(Ω)F+J(v)<0. Thus the minimum point of J is nontrivial.

Theorem 11.

Let (H1)–(H5) hold with p<q<r. Then problem (2) has a nonnegative nontrivial weak solution in W01,p(Ω)F+.

Proof.

Let uW01,p(Ω)F+. Then (41)J(tu)=tppuEp-tqqF(u). Thus J(tu)α>0 for t small and J(tu)<0 for t large.

Let {un}W01,p(Ω)F+ satisfy J(un)c in E and J(un)0 in E*. Then (42)c+1J(un)-1qJ(un),un=(1p-1q)unEp. This shows that unE is bounded in E. Up to a subsequence, we obtain unu in E. Thus (43)Ωa(x)|u|p-2u(un-u)dx+Ωb(x)|u|p-2u(un-u)dx0. It follows from Lemmas 3 and 4 that (44)unuin  Lq(Ω;|f|),Lr(Γ;g).

Hence (45)Ωf(x)|u|q-2u(un-u)dx0,Γg(x)|un|r-2un(un-u)dS0,Bn:=Ωf(x)(|un|q-2un-|u|q-2u)(un-u)dx0,Cn:=Γg(x)(|un|r-2un-|u|r-2u)(un-u)dS0. Then (45) give that J(u),un-u0. Notice that J(un)0; we have (46)J(un)-J(u),un-u=An+Bn-Cn0, where (47)An=Ωa(x)(|un|p-2un-|u|p-2u)(un-u)dx+Ωb(x)(|un|p-2un-|u|p-2u)(un-u)dx.

Using the standard inequality in N given by (48)|ξ|p-2ξ-|η|p-2η,ξ-ηCp|ξ-η|p,  p2,|ξ|p-2ξ-|η|p-2η,ξ-ηCp|ξ-η|2(|ξ|+|η|)p-2,1<p<2, we have from (46) that unu in E. Thus J(·) satisfies (PS) condition. Then the assertion of this theorem follows from Lemma 5.

Next, we seek for a solution in EW01,p(Ω) with p<q<r. In this case, we find it necessary to strengthen our hypothesis by assuming that the function g(·) is positive. That is, (H5) will be replaced by

the function g(x)>0 and g(x)Lσ(Γ) with p/(p-r)σ>r0=p*/(p*-r).

Theorem 12.

Let (H1)–(H4) and (H5) hold with p<q<r. Suppose also (49):={(F(v))r-p>ξ(G(v))q-pvEp(r-q)}, where ξ=(r-p)r-p/((q-p)q-p(r-q)r-q)(r/p)r-q. Then problem (2) has a nonnegative nontrivial weak solution in E/W01,p(Ω).

Proof.

Define (50)v0={v:G(vvE)G(v0v0E)}. Fix v0 with v0. In view of assumption (H5), we have G(v0)>0. For all uv0, there exist t>0 and vv0 with vE=1 such that u=tv. Moreover, G(v)G(v0/v0E)>0. Thus (51)J(u)=tppvEp+trrG(v)-tqqF(v), as t. Hence J(·) is coercive in v0. By Lemmas 3 and 4, it is easy to verify that J is weakly lower semicontinuous. So J has a minimum point u in v0 and u is a weak solution of (2).

In the following, we prove that infuv0J(u)<0. Notice that ψ(t,v)=0 as t0, ψ(t,v)=- as t+, and ψt(t,v)=(q-p)tq-p-1F(v)-(r-p)tr-p-1G(v); we infer that ψ(t,v) attain its maximum at tM(v), where (52)tM(v)=(q-pr-pF(v)G(v))1/(r-q).

If 0<vEp<ψ(tM(v),v), then (13) has exactly two solutions t1(v) and t2(v) with 0<t1(v)<tM(v)<t2(v). Let t(v)=t2(v). We have from (11) and (12) that (53)J(t(v)v)=q-ppq(t(v))pvEp-r-qqr(t(v))rG(v).

Let vv0. It follows that (54)(t(v))r-p>(tM(v))r-p>r(q-p)vEpp(r-q)G(v), which ensures J(t(v)v)<0. Thus infuv0J(u)<0. This implies that the weak solution of (2) is nontrivial.

Remark 13.

Condition (49) may be viewed as grading the “strength” of interaction induced by F(u) and G(u). Hence, qualitatively speaking, one may rephrase Theorem 12 as saying that if p<q<r, then problem (2) admits a nontrivial weak solution provided that F(·) “prevails” over G(·).

4. Multiplicity of Nonnegative Solutions

In this section, we establish multiplicity results for the cases q>max{p,r} and q<min{p,r} by Lemmas 6 and 7, respectively. To this purpose, the assumption (H4) will be replaced by the following:

the function f(x) satisfies f(x)>0 in Ω and f(x)Lδ(Ω) with p/(p-q)δ>q0=p*/(p*-q).

Theorem 14.

Let (H1)–(H3), (H4), and (H5) hold with q>max{p,r}. Then problem (2) has a sequence of solutions uk in E with J(uk) as k.

Proof.

We will prove this theorem by fountain theorem. The proof is divided into three steps.

(1) Let Yk and Zk be defined by (23) and βk=supuZk,uE=1uLq(Ω;f). Then it follows that βk0 (see ). Therefore, we have (55)J(u)1puEp-1qF(u)1puEp-1qβkquEq. Choosing rk=(q/(2pβkq))1/(q-p), we obtain that if uZk,uE=rk, then (56)J(u)12prkp+. Thus (B1) in Lemma 6 is proved.

(2) Since in the finite dimensional space Yk all norms are equivalent, there exist C>0 such that F(u)CuEq hold for all uYk. Thus by (19), (57)J(u)1puEp+1rS2-r/pgLσ(Γ)uEr-1qCuEq,uYk. Therefore (B2) in Lemma 6 is satisfied for every ρk:=uE>0 large enough.

(3) Let {un} be a (PS)c sequence of J. Then we have (58)c+1J(un)-1qJ(un),un=(1p-1q)unEp+(1r-1q)G(u)(1p-1q)unEp. Therefore {un} is bounded in E. Similar to the proof of Theorem 11, we can verify that J(·) satisfies (PS) condition.

Obviously, J(·) is an even functional and J(0)=0. Thus the assertion of Theorem 14 follows from Lemma 6.

Theorem 15.

Let (H1)–(H3), (H4), and (H5) hold with q<min{p,r}. Then problem (2) has a sequence of solutions uk in E with J(uk)<0 and J(uk)0 as k.

Proof.

Since (59)J(u)=1puEp+1rG(u)-1qF(u)1puEp-1qS1-q/pfLδ(Ω)uEq and q<p,  J(·) is coercive and bounded from below. As before, we can verify that (PS) condition holds.

Let Yk be defined by (23) and uYk. Since Yk is a finite dimensional space and q<min{p,r}, we can choose ρk>0 small enough such that (60)supuYk,uE=ρkJ(u)1puEp+1rS2-r/pgLσ(Γ)uEr-1qCuEq<0. We obtain a sequence of solutions uk by Lemma 7.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (2013B09914).

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