IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi Publishing Corporation67352610.1155/2010/673526673526Research ArticleMultiple Solutions of Quasilinear Elliptic Equations in NLinHuei-liSchechterMartin D.Department of Natural SciencesCenter for General EducationChang Gung UniversityTaoyuan 333Taiwancgu.edu.tw20100803201020100110200915012010010320102010Copyright © 2010This 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.

Assume that Q is a positive continuous function in N and satisfies some suitable conditions. We prove that the quasilinear elliptic equation -Δpu+|u|p-2u=Q(z)|u|q-2u in N admits at least two solutions in N (one is a positive ground-state solution and the other is a sign-changing solution).

1. Introduction

For N3, 2p<N, and p<q<p*=Np/(N-p), we consider the quasilinear elliptic equations

-Δpu+|u|p-2u=Q(z)|u|q-2uin  N,uW1,p(N),-Δpu+|u|p-2u=Q|u|q-2uin  N,uW1,p(N), where Δp is the p-Laplacian operator, that is,

Δpu=i=1Nzi(|u|p-2uzi). Let Q be a positive continuous function in N and satisfy

Q(z)Q=lim|z|Q(z)>0,Q(z)>Qon  a  set  of  positive  measure. Associated with (1.1) and (1.2), we define the functionals a,b,b,J, and J, for uW1,p(N),

a(u)=N(|u|p+|u|p)dz=u1,pp,b(u)=NQ(z)|u|qdz,b(u)=NQ|u|qdz,J(u)=1pa(u)-1qb(u),J(u)=1pa(u)-1qb(u). It is easy to verify that the functionals a,b,b,J, and J are C1.

For the case p=2, Lions [1, 2] proved that if lim|z|Q(z)=Q, and Q(z)Q>0, then (1.1) has a positive ground-state solution in N. Benci and Cerami  proved that (1.2) does not have any ground-state solution in an exterior domain. Bahri and Li  proved that there is at least one positive solution of (1.1) in N (or an exterior domain) when lim|z|Q(z)=Q>0 and Q(z)Q-Cexp(-δ|z|) for δ>2. Cao  has studied the multiplicity of solutions (one is a positive ground-state solution and the other is a nodal solution) of (1.1) with Neumann condition in an exterior domain as follows. Assume that lim|z|Q(z)=Q>0, and Q(z)Q+C|z|-mexp(-δ|z|) for C>0, m<(N-1)/2, δ=q/(q+1), then (1.1) has at least two nontrivial solutions (one is a positive ground-state solution and the other is a nodal solution) in an exterior domain.

This article is motivated by the above papers. If Q is a positive continuous function in N and satisfies (Q1), then we prove that (1.1) admits a positive ground-state solution in N. Combine it with some ideas of Cerami et al.  to show that if Q also satisfies Q(z)Q+Cexp(-δ|z|) for 0<δ<θ=(p-1)-1/p, then a nodal solution of (1.1) exists.

2. Preliminaries

We define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in W1,p(N) for J as follows.

Definition 2.1.

(i) For β, a sequence {un} is a (PS)β-sequence in W1,p(N) for J if J(un)=β+on(1) and J(un)=on(1) strongly in W-1,p(N) as n, where W-1,p(N) is the dual space of W1,p(N) and 1/p+1/p=1

(ii) J satisfies the (PS)β-condition in W1,p(N) if every (PS)β-sequence in W1,p(N) for J contains a convergent subsequence.

Lemma 2.2.

Let β and let {un} be a (PS)β-sequence in W1,p(N) for J, then {un} is a bounded sequence in W1,p(N). Moreover, a(un)=b(un)+on(1)=(qp/(q-p))β+on(1) as n and β0.

Proof.

Since p2, we have that a(un)p1 if a(un)1 and a(un)pa(un) if a(un)>1. For sufficiently large n, we get |β|+2+a(un)|β|+1+a(un)pJ(un)-1qJ(un),un=(1p-1q)a(un). It follows that {un} is bounded in W1,p(N). Then J(un),un=on(1) as n. Thus, β+on(1)=J(un)=(1p-1q)a(un)+on(1)=(1p-1q)b(un)+on(1), that is, a(un)=b(un)+on(1)=(qp/(q-p))β+on(1) as n and β0.

Define

α(N)=infuM(N)J(u), where M(N)={uW1,p(N){0}a(u)=b(u)}, and

α(N)=infuM(N)J(u), where M(N)={uW1,p(N){0}a(u)=b(u)}.

Lemma 2.3.

Let u be a sign-changing solution of (1.1). Then J(u)2α(N).

Proof.

Define u+=max{u,0} and u-=max{-u,0}. Since u is a sign-changing solution of (1.1), then u- is nonnegative and nonzero. Multiply (1.1) by u- and integrate it to obtain N(|u|p-2uu-+|u|p-2uu-)dz=NQ(z)|u|q-2uu-dz, that is, u-M(N) and J(u-)α(Ω). Similarly, J(u+)α(N). Hence, J(u)=J(u+)+J(u-)2α(N).

Lemma 2.4.

(i) For each uW1,p(N){0}, there exists a positive number su such that suuM(N) and sups0J(su)=J(suu).

(ii) Let β>0 and let {un} be a sequence in W1,p(N){0} for J such that a(un)=b(un)+o(1) and J(un)=β+o(1). Then there is a sequence {sn} in + such that sn=1+o(1),{snun}M(N), and J(snun)=β+o(1) as n.

Proof.

(i) For each uW01,p(N){0} and s0, let hu(s)=J(su)=sppa(u)-sqqb(u). Thus, hu(s)=sp-1a(u)-sq-1b(u). Define su=(a(u)/b(u))1/(q-p)>0, then hu(su)=0, that is, suuM(N).

(ii) By (i), there exists a sequence {sn} in + such that {snun}M(N), that is, snpa(un)=snqb(un) for each n. Since a(un)=b(un)+o(1) and J(un)=β+o(1), we have that sn=1+o(1). Hence, J(snun)=β+o(1) as n.

Lemma 2.5.

There exists c>0 such that u1,pc>0 for each uM(N), where c is independent of u.

Proof.

For each uM(N), by the Sobolev inequality, we obtain that u1,pp=NQ(z)|u|qdzc1u1,pq. This implies that u1,pc1-1/(q-p)=c>0 for each uM(N).

By Lemma 2.5, α(N)>0.

Lemma 2.6.

Let uM(N) such that J(u)=minvM(N)J(v)=α(N), then u is a nonzero solution of (1.1) in N.

Proof.

Suppose that ψ(v)=N(|v|p+|v|p)dz-NQ(z)|v|qdz, then ψ(v),v=(p-q)N(|v|p+|v|p)dz<0for  each  vM(N). Since J(u)=minvM(N)J(v), by the Lagrange multiplier theorem, there is a λ such that J(u)=λψ(u) in W-1,p(N). Then we have 0=J(u),u=λψ(u),u. Thus, λ=0 and J(u)=0 in W-1,p(N). Therefore, u is a nonzero solution of (1.1) in N with J(u)=α(N).

Lemma 2.7.

There is a (PS)α(N)-sequence in W1,p(N) for J.

Proof.

Let {un}M(N) be a minimizing sequence of α(N). Applying the Ekeland principle, there exists a sequence {vn}M(N) such that vn-un1,p<1/n, J(vn)=α(N)+o(1), and J|M(N)(vn)=o(1) strongly in W-1,p(N) as n. Let ψ(u)=a(u)-b(u) for each uW1,p(N){0}, then M(N)={uW1,p(N){0}ψ(u)=0}. Thus, there exists a sequence {θn} such that J(vn)=θnψ(vn)+on(1), where on(1)0 as n. Since vnM(N), we have that 0=J(vn),vn=θnψ(vn),vn+on(1),vn,ψ(vn),vn=(p-q)a(vn)0n. Hence, θn0 as n. This implies that J(vn)=o(1) strongly in W-1,p(N) as n, that is, {vn}M(N) is a (PS)α(Ω)-sequence in W1,p(N) for J.

Remark 2.8.

The above definitions and lemmas also hold for J,M(N), and α(N).

3. Existence of a Ground-State Solution

Using the arguments by Lions [1, 2], Benci and Cerami , Struwe , and Alves , we have the following decomposition lemma.

Lemma 3.1 (Palais-Smale Decomposition Lemma for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M205"><mml:mrow><mml:mi>J</mml:mi></mml:mrow></mml:math></inline-formula>).

Assume that Q is a positive continuous function in N and lim|z|Q(z)=Q>0. Let {un} be a (PS)β-sequence in W1,p(N) for J. Then there are a subsequence {un}, a positive integer l, sequences {zni}n=1 in N, functions u in W1,p(N), and wi0 in W1,p(N) for 1il such that |zni|for  1il,-Δpu+|u|p-2u=Q(z)|u|q-2uin  N,-Δpwi+|wi|p-2wi=Q|wi|q-2wiin  N,un=u+i=1lwi(·-zni)+on(1)strongly  in  W1,p(N),J(un)=J(u)+i=1lJ(wi)+on(1). In addition, if un0, then u0 and wi0 for 1il.

Lemma 3.2.

Let {un}M(N) be a (PS)β-sequence in W1,p(N) for J with 0<β<α(N). Then there exist a subsequence {un} and a nonzero uW1,p(N) such that unu strongly in W1,p(N) and J(u)=β, that is, J satisfies the (PS)β-condition in W1,p(N).

Proof.

Since {un}M(N) is a (PS)β-sequence in W1,p(N) for J with 0<β<α(N), by Lemma 2.2, {un} is bounded in W1,p(N). Thus, there exist a subsequence {un} and uW1,p(N) such that unu weakly in W1,p(N). It is easy to check that u is a solution of (1.1) in N. Applying Palais-Smale Decomposition Lemma 3.1, we get α>β=J(un)lα. Then l=0 and u0. Hence, unu strongly in W1,p(N) and J(u)=β.

Let wW1,p(N) be the positive ground-state solution of (1.2) in N. Using the same arguments by Li and Yan  and Marcos do Ó [10, Lemma 3.8], or see Serrin and Tang [11, page 899] and Li and Zhao [12, Theorem 1.1], we obtain the following results:

wL(N)Cloc1,γ0(N) for some 0<γ0<1 and lim|z|w(z)=0;

for any ɛ>0, there exist positive numbers C1 and C2 such that C2exp(-(θ+ɛ)|z|)w(z)C1exp(-(θ-ɛ)|z|)zN, where θ=(p-1)-1/p.

Remark 3.3.

Similarly, we also show that all positive solutions of (1.1) in N have exponential decay.

By Lemma 3.2, we can prove the following theorem.

Theorem 3.4.

Assume that Q is a positive continuous function in N and satisfies (Q1). Then there exists a positive ground-state solution u0 of (1.1) in N.

Proof.

Let wW1,p(N) be the positive ground-state solution of (1.2) in N, then w is a minimizer of α(N) and N(|w|p+wp)dz=NQwqdz. By Lemma 2.4(i), there exists a positive number sw such that swwM(N), that is, N(|(sww)|p+(sww)p)dz=NQ(z)(sww)qdz. Since Q(z)>Q on a set of positive measure, we can deduce that sw<1. Therefore, α(N)J(sww)=(1p-1q)(sw)pN(|w|p+wp)dz<(1p-1q)N(|w|p+wp)dz=(1p-1q)NQwqdz=α(N). Applying Lemma 3.2, there exists u0W1,p(N) such that J(u0)=α(N). From the results of Lemmas 2.6 and 2.3, by Maximum Principle, u0 is a positive ground-state solution of (1.1) in N.

4. Existence of a Nodal Solution

In this section, assume that Q is a positive continuous function in N and satisfies (Q1). In order to prove Lemma 4.8, Q also satisfies the following condition (Q2): there exist some constants C>0 and 0<δ<θ=(p-1)-1/p such that

Q(z)Q+Cexp(-δ|z|)zN. Let h be a functional in W1,p(N) defined by

h(u)={b(u)a(u)for  u0,0for  u=0. We define

M0={uW1,p(N)h(u+)=1,h(u-)=1}M(N),𝒩={uW1,p(N)|h(u±)-1|<12}M0, where u+=max{u,0} and u-=max{-u,0}.

Lemma 4.1.

(i) If uW1,p(N) changes sign, then there exist positive numbers s±(u)=s± such that s+u+M(N) and s-u-M(N).

(ii) There exists c>0 such that u±1,pc>0 for each u𝒩.

Proof.

(i) Since u+ and u- are nonzero and nonnegative, by Lemma 2.4(i), it is easy to obtain the result.

(ii) For each u𝒩, by Lemma 2.4(i), there exists s±(u)=s±>0 such that s±u±M(N). Then 12<(s±)p-q=b(u±)a(u±)<32for  each  u𝒩. By Lemma 2.5, we have s±u±1,pcfor  some  c>0and  each  u𝒩. Hence, u±1,pc/s±c>0 for each u𝒩.

Consider these minimization problem

γ(N)=infuM0J(u). By Lemma 4.1, γ(N)>0.

Lemma 4.2.

There exists a sequence {un}𝒩 such that J(un)=γ(N)+on(1) and J(un)=on(1) strongly in W-1,p(N) as n.

Proof.

It is similar to the proof of Zhu .

Lemma 4.3.

Let f and g be real-valued functions in N. If g(z)>0 in N, then one has the following inequalities:

(f+g)+f+,

(f+g)-f-,

(f-g)+f+,

(f-g)-f-.

Lemma 4.4.

Let {un}𝒩 be a (PS)γ(N)-sequence in W1p(N) for J satisfying α(N)<γ(N)<α(N)+α(N)(<2α(N)). Then there exists u*M0 such that un converges to u* strongly in W1,p(N) and u* is a higher-energy solution of (1.1) such that J(u*)=γ(N).

Proof.

By the definition of the (PS)γ(N)-sequence in W1,p(N) for J, it is easy to see that {un} is a bounded sequence in W1,p(N) and satisfies N(|un±|p+|un±|p)dz=NQ(z)|un±|qdz+on(1). By Lemma 4.1(ii), there exists c>0 such that cN(|un±|p+|un±|p)dz=NQ(z)|un±|qdz+on(1). Using the Palais-Smale Decomposition Lemma 3.1, then we have γ(N)=J(u*)+i=1lJ(wi), where u* is a solution of (1.1) in N and wi is a solution of (1.2) in N. Since J(wi)α(N) for each i and α(N)<α(N), we have l1. Now we want to show that l=0. On the contrary, suppose that l=1.

w1 is a sign-changing solution of (1.2): by Lemma 2.3 and Remark 2.8, we have γ(N)2α(N), which is a contradiction.

w1 is a constant-sign solution of (1.2): we may assume that w1>0. Applying the Decomposition Lemma 3.1, there exists a sequence {zn1} in N such that |zn1|, and un-u*-w1(·-zn1)1,p=on(1).

By the Sobolev continuous embedding inequality, we obtain un-u*-w1(·-zn1)Lq=on(1). Since w1>0, by Lemma 4.3, then (un-u*)-Lq=on(1)as  n.

Suppose that u*0; we obtain un-Lq=on(1) as n. Then 0<cNQ(z)|un-|qdz=on(1), which is a contradiction.

Suppose that u*0. We have γ(N)=J(u*)+J(w1)α(N)+α(N), which is a contradiction.

By (i) and (ii), then l=0. Thus, un-u*1,p=on(1) as n and J(u*)=γ(N). Finally, we claim that u* is a sign-changing solution of (1.1) in N. If u*>0(or<0), by Lemma 4.3, then un-Lq=on(1)(or  un-Lq=on(1)). Similarly, we have the inequality (4.12), which is a contradiction. Moreover, by Lemma 2.3, 2α(N)γ(N).

Recall that w is the positive ground-state solution of (1.2) in N. For any ɛ>0, there exist positive numbers C1 and C2 such that

C2exp(-(θ+ɛ)|z|)w(z)C1exp(-(θ-ɛ)|z|)zN, where θ=(p-1)-1/p. Define

wn(z)=w(z-zn)where  zn=(0,,0,n)N. Clearly, wn(z)W1,p(N).

Lemma 4.5.

There are n0 and real numbers t1* and t2* such that for nn0t1*u0-t2*wnM0,γ(N)J(t1*u0-t2*wn), where 1/pt1*, t2*p, and u0 is the positive ground-state solution of (1.1) in N.

Proof.

Applying the mean value theorem by Miranda , the proof is similar to that of Zhu  (or see Hsu [15, page 728]).

We need the following lemmas to prove that sup1/pt1*,t2*pJ(t1*u0-t2*wn)<α(N)+α(N) for sufficiently large n.

Lemma 4.6.

Let E be a domain in N. If f:E satisfies E|f(z)eσ|z||dz<for  some  σ>0, then (Ef(z)e-σ|z-z¯|dz)eσ|z¯|=Ef(z)eσz,z¯/|z¯|dz+o(1)as  |z¯|.

Proof.

Since σ|z¯|σ|z|+σ|z-z¯|, we have |f(z)e-σ|z-z¯|eσ|z¯|||f(z)eσ|z||. Since -σ|z-z¯|+σ|z¯|=σ(z,z¯/|z¯|)+o(1) as |z¯|, then the lemma follows from the Lebesgue-dominated convergence theorem.

Lemma 4.7.

For all x,yN, one has the following inequality: |x-y|ρ(|x|ρ-2x-|y|ρ-2y)(x-y),where  ρ2.

Proof.

See Yang [16, Lemma 4.2.].

Lemma 4.8.

There exists an n0* such that for nn0*n0γ(N)sup1/pt1*,t2*pJ(t1*u0-t2*wn)<α(N)+α(N), where u0 is a positive ground-state solution of (1.1) in N.

Proof.

By Lemma 4.7, then J(t1*u0-t2*wn)=1pt1*u0-t2*wn1,pp-1qb(t1*u0-t2*wn)1p{N(|(t1*u0)|p-2(t1*u0)-|(t2*wn)|p-2(t2*wn))((t1*u0)-(t2*wn))}+1p{N(|t1*u0|p-2(t1*u0)-|t2*wn|p-2(t2*wn))(t1*u0-t2*wn)}-1qb(t1*u0-t2*wn)J(t1*u0)+J(t2*w)-(t2*)qqN(Q(z)-Q)w(z-zn)qdz-1qb(t1*u0-t2*wn)+1qb(t1*u0)+1qb(t2*wn). Since supt0J(tu0)=α(N) and supt0J(tw)=α(N), using the inequality |c1-c2|q>c1q+c2q-K(c1q-1c2+c1c2q-1), for any c1,c2>0, and some positive constant K, then sup1/pt1*,t2*pJ(t1*u0-t2*wn)α(N)+α(N)-1pqqN(Q(z)-Q)w(z-zn)qdz+K[N(u0q-1wn+wnq-1u0)dz].

Since Q(z)Q+Cexp(-δ|z|) for some constants C>0 and 0<δ<θ, by Lemma 4.6, we have that there exists an n1n0 such that for nn1N(Q(z)-Q)w(z-zn)qdzCexp(-min{δ,q(θ+ɛ)}|z¯|)Cexp(-δn).

Applying Lemma 4.6, there exists an n2n1 such that for nn2Nu0q-1wndzC1Nexp(-(q-1)(θ-ɛ)|z|)exp(-(θ-ɛ)|z-zn|)dzC1′′exp(-(θ-ɛ)n).

Similarly, we also obtain that there exists an n3n2 such that for nn3Nwnq-1u0dzC1′′′exp(-(θ-ɛ)n). By (i) and (ii), choosing 0<ɛ<θ-δ, we can find an n0*n3n0 such that for nn0*sup1/pt1,t2pJ(t1*u0-t2*wn)<α(N)+α(N).

Theorem 4.9.

Assume that Q is a positive continuous function in N and satisfies (Q1) and (Q2), then (1.1) has a positive solution and a nodal solution in N.

Proof.

By Lemmas 4.2, 4.4, 4.5, and 4.8 and Theorem 3.4, we obtain the proof.

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