# Multiple Solutions of Quasilinear Elliptic Equations inℝN

Authors: Huei-li Lin
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

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


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## Body

## 1. Introduction

ForN≥3, 2≤p<N, and p<q<p*=Np/(N-p), we consider the quasilinear elliptic equations(1.1)-Δpu+|u|p-2u=Q(z)|u|q-2uinℝN,u∈W1,p(ℝN),(1.2)-Δpu+|u|p-2u=Q∞|u|q-2uinℝN,u∈W1,p(ℝN),
where Δp is the p-Laplacian operator, that is,(1.3)Δpu=∑i=1N∂∂zi(|∇u|p-2∂u∂zi).
Let Q be a positive continuous function in ℝN and satisfy(Q1)Q(z)≥Q∞=lim|z|→∞Q(z)>0,Q(z)>Q∞onasetofpositivemeasure.
Associated with (1.1) and (1.2), we define the functionals a,b,b∞,J, and J∞, for u∈W1,p(ℝN),(1.4)a(u)=∫ℝN(|∇u|p+|u|p)dz=∥u∥1,pp,b(u)=∫ℝNQ(z)|u|qdz,b∞(u)=∫ℝNQ∞|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 casep=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 ℝN. Benci and Cerami [3] proved that (1.2) does not have any ground-state solution in an exterior domain. Bahri and Li [4] proved that there is at least one positive solution of (1.1) in ℝN (or an exterior domain) when lim|z|→∞Q(z)=Q∞>0 and Q(z)≥Q∞-Cexp(-δ|z|) for δ>2. Cao [5] 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. IfQ is a positive continuous function in ℝN and satisfies (Q1), then we prove that (1.1) admits a positive ground-state solution in ℝN. Combine it with some ideas of Cerami et al. [6] 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 inW1,p(ℝN) for J as follows.Definition 2.1.
(i) Forβ∈ℝ, a sequence {un} is a (PS)β-sequence in W1,p(ℝN) for J if J(un)=β+on(1) and J′(un)=on(1) strongly in W-1,p′(ℝN) as n→∞, where W-1,p′(ℝN) is the dual space of W1,p(ℝN) and 1/p+1/p′=1
(ii)J satisfies the (PS)β-condition in W1,p(ℝN) if every (PS)β-sequence in W1,p(ℝN) for J contains a convergent subsequence.Lemma 2.2.
Letβ∈ℝ and let {un} be a (PS)β-sequence in W1,p(ℝN) for J, then {un} is a bounded sequence in W1,p(ℝN). Moreover, a(un)=b(un)+on(1)=(qp/(q-p))β+on(1) as n→∞ and β≥0.Proof.
Sincep≥2, we have that a(un)p≤1 if a(un)≤1 and a(un)p≤a(un) if a(un)>1. For sufficiently large n, we get
(2.1)|β|+2+a(un)≥|β|+1+a(un)p≥J(un)-1q〈J′(un),un〉=(1p-1q)a(un).
It follows that {un} is bounded in W1,p(ℝN). Then 〈J′(un),un〉=on(1) as n→∞. Thus,
(2.2)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 β≥0.Define(2.3)α(ℝN)=infu∈M(ℝN)J(u),
where M(ℝN)={u∈W1,p(ℝN)∖{0}∣a(u)=b(u)}, and(2.4)α∞(ℝN)=infu∈M∞(ℝN)J∞(u),
where M∞(ℝN)={u∈W1,p(ℝN)∖{0}∣a(u)=b∞(u)}.Lemma 2.3.
Letu be a sign-changing solution of (1.1). Then J(u)≥2α(ℝN).Proof.
Defineu+=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
(2.5)∫ℝN(|∇u|p-2∇u∇u-+|u|p-2uu-)dz=∫ℝNQ(z)|u|q-2uu-dz,
that is, u-∈M(ℝN) and J(u-)≥α(Ω). Similarly, J(u+)≥α(ℝN). Hence,
(2.6)J(u)=J(u+)+J(u-)≥2α(ℝN).Lemma 2.4.
(i) For eachu∈W1,p(ℝN)∖{0}, there exists a positive number su such that suu∈M(ℝN) and sups≥0J(su)=J(suu).
(ii) Letβ>0 and let {un} be a sequence in W1,p(ℝN)∖{0} for J such that a(un)=b(un)+o(1) and J(un)=β+o(1). Then there is a sequence {sn} in ℝ+ such that sn=1+o(1),{snun}⊂M(ℝN), and J(snun)=β+o(1) as n→∞.Proof.
(i) For eachu∈W01,p(ℝN)∖{0} and s≥0, let
(2.7)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, suu∈M(ℝN).
(ii) By (i), there exists a sequence{sn} in ℝ+ such that {snun}⊂M(ℝN), 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 existsc>0 such that ∥u∥1,p≥c>0 for each u∈M(ℝN), where c is independent of u.Proof.
For eachu∈M(ℝN), by the Sobolev inequality, we obtain that
(2.8)∥u∥1,pp=∫ℝNQ(z)|u|qdz≤c1∥u∥1,pq.
This implies that ∥u∥1,p≥c1-1/(q-p)=c>0 for each u∈M(ℝN).By Lemma2.5, α(ℝN)>0.Lemma 2.6.
Letu∈M(ℝN) such that
(2.9)J(u)=minv∈M(ℝN)J(v)=α(ℝN),
then u is a nonzero solution of (1.1) in ℝN.Proof.
Suppose thatψ(v)=∫ℝN(|∇v|p+|v|p)dz-∫ℝNQ(z)|v|qdz, then
(2.10)〈ψ′(v),v〉=(p-q)∫ℝN(|∇v|p+|v|p)dz<0foreachv∈M(ℝN).
Since J(u)=minv∈M(ℝN)J(v), by the Lagrange multiplier theorem, there is a λ∈ℝ such that J′(u)=λψ′(u) in W-1,p′(ℝN). Then we have
(2.11)0=〈J′(u),u〉=λ〈ψ′(u),u〉.
Thus, λ=0 and J′(u)=0 in W-1,p′(ℝN). Therefore, u is a nonzero solution of (1.1) in ℝN with J(u)=α(ℝN).Lemma 2.7.
There is a(PS)α(ℝN)-sequence in W1,p(ℝN) for J.Proof.
Let{un}⊂M(ℝN) be a minimizing sequence of α(ℝN). Applying the Ekeland principle, there exists a sequence {vn}⊂M(ℝN) such that ∥vn-un∥1,p<1/n, J(vn)=α(ℝN)+o(1), and J′|M(ℝN)(vn)=o(1) strongly in W-1,p′(ℝN) as n→∞. Let ψ(u)=a(u)-b(u) for each u∈W1,p(ℝN)∖{0}, then
(2.12)M(ℝN)={u∈W1,p(ℝN)∖{0}∣ψ(u)=0}.
Thus, there exists a sequence {θn}⊂ℝ such that J′(vn)=θnψ′(vn)+on(1), where on(1)→0 as n→∞. Since vn∈M(ℝN), we have that
(2.13)0=〈J′(vn),vn〉=θn〈ψ′(vn),vn〉+〈on(1),vn〉,〈ψ′(vn),vn〉=(p-q)a(vn)≠0∀n.
Hence, θn→0 as n→∞. This implies that J′(vn)=o(1) strongly in W-1,p′(ℝN) as n→∞, that is, {vn}⊂M(ℝN) is a (PS)α(Ω)-sequence in W1,p(ℝN) for J.Remark 2.8.
The above definitions and lemmas also hold forJ∞,M∞(ℝN), and α∞(ℝN).


## 3. Existence of a Ground-State Solution

Using the arguments by Lions [1, 2], Benci  and Cerami [3], Struwe [7], and Alves [8], we have the following decomposition lemma.Lemma 3.1 (Palais-Smale Decomposition Lemma forJ).
Assume thatQ is a positive continuous function in ℝN and lim|z|→∞Q(z)=Q∞>0. Let {un} be a (PS)β-sequence in W1,p(ℝN) for J. Then there are a subsequence {un}, a positive integer l, sequences {zni}n=1∞ in ℝN, functions u in W1,p(ℝN), and wi≠0 in W1,p(ℝN) for 1≤i≤l such that
(3.1)|zni|→∞for1≤i≤l,-Δpu+|u|p-2u=Q(z)|u|q-2uinℝN,-Δpwi+|wi|p-2wi=Q∞|wi|q-2wiinℝN,un=u+∑i=1lwi(·-zni)+on(1)stronglyinW1,p(ℝN),J(un)=J(u)+∑i=1lJ∞(wi)+on(1).
In addition, if un≥0, then u≥0 and wi≥0 for 1≤i≤l.Lemma 3.2.
Let{un}⊂M(ℝN) be a (PS)β-sequence in W1,p(ℝN) for J with 0<β<α∞(ℝN). Then there exist a subsequence {un} and a nonzero u∈W1,p(ℝN) such that un→u strongly in W1,p(ℝN) and J(u)=β, that is, J satisfies the (PS)β-condition in W1,p(ℝN).Proof.
Since{un}⊂M(ℝN) is a (PS)β-sequence in W1,p(ℝN) for J with 0<β<α∞(ℝN), by Lemma 2.2, {un} is bounded in W1,p(ℝN). Thus, there exist a subsequence {un} and u∈W1,p(ℝN) such that un⇀u weakly in W1,p(ℝN). It is easy to check that u is a solution of (1.1) in ℝN. Applying Palais-Smale Decomposition Lemma 3.1, we get
(3.2)α∞>β=J(un)≥lα∞.
Then l=0 and u≠0. Hence, un→u strongly in W1,p(ℝN) and J(u)=β.Letw∈W1,p(ℝN) be the positive ground-state solution of (1.2) in ℝN. Using the same arguments by Li and Yan [9] 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:(i)
w∈L∞(ℝN)∩Cloc1,γ0(ℝN) for some 0<γ0<1 and lim|z|→∞w(z)=0;(ii)
for anyɛ>0, there exist positive numbers C1 and C2 such that
(3.3)C2exp(-(θ+ɛ)|z|)≤w(z)≤C1exp(-(θ-ɛ)|z|)∀z∈ℝN,
where θ=(p-1)-1/p.Remark 3.3.
Similarly, we also show that all positive solutions of (1.1) in ℝN have exponential decay.By Lemma3.2, we can prove the following theorem.Theorem 3.4.
Assume thatQ is a positive continuous function in ℝN and satisfies (Q1). Then there exists a positive ground-state solution u0 of (1.1) in ℝN.Proof.
Letw∈W1,p(ℝN) be the positive ground-state solution of (1.2) in ℝN, then w is a minimizer of α∞(ℝN) and
(3.4)∫ℝN(|∇w|p+wp)dz=∫ℝNQ∞wqdz.
By Lemma 2.4(i), there exists a positive number sw such that sww∈M(ℝN), that is, ∫ℝN(|∇(sww)|p+(sww)p)dz=∫ℝNQ(z)(sww)qdz. Since Q(z)>Q∞ on a set of positive measure, we can deduce that sw<1. Therefore,
(3.5)α(ℝN)≤J(sww)=(1p-1q)(sw)p∫ℝN(|∇w|p+wp)dz<(1p-1q)∫ℝN(|∇w|p+wp)dz=(1p-1q)∫ℝNQ∞wqdz=α∞(ℝN).
Applying Lemma 3.2, there exists u0∈W1,p(ℝN) such that J(u0)=α(ℝN). From the results of Lemmas 2.6 and 2.3, by Maximum Principle, u0 is a positive ground-state solution of (1.1) in ℝN.


## 4. Existence of a Nodal Solution

In this section, assume thatQ is a positive continuous function in ℝN 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(Q2)Q(z)≥Q∞+Cexp(-δ|z|)∀z∈ℝN.
Let h be a functional in W1,p(ℝN) defined by(4.1)h(u)={b(u)a(u)foru≠0,0foru=0.
We define(4.2)M0={u∈W1,p(ℝN)∣h(u+)=1,h(u-)=1}⊂M(ℝN),𝒩={u∈W1,p(ℝN)∣|h(u±)-1|<12}⊃M0,
where u+=max{u,0} and u-=max{-u,0}.Lemma 4.1.
(i) Ifu∈W1,p(ℝN) changes sign, then there exist positive numbers s±(u)=s± such that s+u+∈M(ℝN) and s-u-∈M(ℝN).
(ii) There existsc′>0 such that ∥u±∥1,p≥c′>0 for each u∈𝒩.Proof.
(i) Sinceu+ and u- are nonzero and nonnegative, by Lemma 2.4(i), it is easy to obtain the result.
(ii) For eachu∈𝒩, by Lemma 2.4(i), there exists s±(u)=s±>0 such that s±u±∈M(ℝN). Then
(4.3)12<(s±)p-q=b(u±)a(u±)<32foreachu∈𝒩.
By Lemma 2.5, we have
(4.4)∥s±u±∥1,p≥cforsomec>0andeachu∈𝒩.
Hence, ∥u±∥1,p≥c/s±≥c′>0 for each u∈𝒩.Consider these minimization problem(4.5)γ(ℝN)=infu∈M0J(u).
By Lemma 4.1, γ(ℝN)>0.Lemma 4.2.
There exists a sequence{un}⊂𝒩 such that J(un)=γ(ℝN)+on(1) and J′(un)=on(1) strongly in W-1,p(ℝN) as n→∞.Proof.
It is similar to the proof of Zhu [13].Lemma 4.3.
Letf and g be real-valued functions in ℝN. If g(z)>0 in ℝN, then one has the following inequalities:(i)
(f+g)+≥f+,(ii)
(f+g)-≤f-,(iii)
(f-g)+≤f+,(iv)
(f-g)-≥f-.Lemma 4.4.
Let{un}⊂𝒩 be a (PS)γ(ℝN)-sequence in W1p(ℝN) for J satisfying
(4.6)α(ℝN)<γ(ℝN)<α(ℝN)+α∞(ℝN)(<2α∞(ℝN)).
Then there exists u*∈M0 such that un converges to u* strongly in W1,p(ℝN) and u* is a higher-energy solution of (1.1) such that J(u*)=γ(ℝN).Proof.
By the definition of the(PS)γ(ℝN)-sequence in W1,p(ℝN) for J, it is easy to see that {un} is a bounded sequence in W1,p(ℝN) and satisfies
(4.7)∫ℝN(|∇un±|p+|un±|p)dz=∫ℝNQ(z)|un±|qdz+on(1).
By Lemma 4.1(ii), there exists c′>0 such that
(4.8)c′≤∫ℝN(|∇un±|p+|un±|p)dz=∫ℝNQ(z)|un±|qdz+on(1).
Using the Palais-Smale Decomposition Lemma 3.1, then we have γ(ℝN)=J(u*)+∑i=1lJ∞(wi), where u* is a solution of (1.1) in ℝN and wi is a solution of (1.2) in ℝN. Since J∞(wi)≥α∞(ℝN) for each i∈ℕ and α(ℝN)<α∞(ℝN), we have l≤1. Now we want to show that l=0. On the contrary, suppose that l=1.(i)
w1 is a sign-changing solution of (1.2): by Lemma 2.3 and Remark 2.8, we have γ(ℝN)≥2α∞(ℝN), which is a contradiction.(ii)
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 ℝN such that |zn1|→∞, and
(4.9)∥un-u*-w1(·-zn1)∥1,p=on(1).By the Sobolev continuous embedding inequality, we obtain
(4.10)∥un-u*-w1(·-zn1)∥Lq=on(1).
Since w1>0, by Lemma 4.3, then
(4.11)∥(un-u*)-∥Lq=on(1)asn→∞.(a)
Suppose thatu*≡0; we obtain ∥un-∥Lq=on(1) as n→∞. Then
(4.12)0<c′≤∫ℝNQ(z)|un-|qdz=on(1),
which is a contradiction.(b)
Suppose thatu*≢0. We have γ(ℝN)=J(u*)+J∞(w1)≥α(ℝN)+α∞(ℝN), which is a contradiction.
By (i) and (ii), then l=0. Thus, ∥un-u*∥1,p=on(1) as n→∞ and J(u*)=γ(ℝN). Finally, we claim that u* is a sign-changing solution of (1.1) in ℝN. 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(ℝN)≤γ(ℝN).Recall thatw is the positive ground-state solution of (1.2) in ℝN. For any ɛ>0, there exist positive numbers C1 and C2 such that(4.13)C2exp(-(θ+ɛ)|z|)≤w(z)≤C1exp(-(θ-ɛ)|z|)∀z∈ℝN,
where θ=(p-1)-1/p. Define(4.14)wn(z)=w(z-zn)wherezn=(0,…,0,n)∈ℝN.
Clearly, wn(z)∈W1,p(ℝN).Lemma 4.5.
There aren0∈ℕ and real numbers t1* and t2* such that for n≥n0(4.15)t1*u0-t2*wn∈M0,γ(ℝN)≤J(t1*u0-t2*wn),
where 1/p≤t1*, t2*≤p, and u0 is the positive ground-state solution of (1.1) in ℝN.Proof.
Applying the mean value theorem by Miranda [14], the proof is similar to that of Zhu [13] (or see Hsu [15, page 728]).We need the following lemmas to prove thatsup1/p≤t1*,t2*≤pJ(t1*u0-t2*wn)<α(ℝN)+α∞(ℝN) for sufficiently large n.Lemma 4.6.
LetE be a domain in ℝN. If f:E→ℝ satisfies
(4.16)∫E|f(z)eσ|z||dz<∞forsomeσ>0,
then
(4.17)(∫Ef(z)e-σ|z-z¯|dz)eσ|z¯|=∫Ef(z)eσ〈z,z¯〉/|z¯|dz+o(1)as|z¯|→∞.Proof.
Sinceσ|z¯|≤σ|z|+σ|z-z¯|, we have
(4.18)|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 allx,y∈ℝN, one has the following inequality:
(4.19)|x-y|ρ≤(|x|ρ-2x-|y|ρ-2y)(x-y),whereρ≥2.Proof.
See Yang [16, Lemma 4.2.].Lemma 4.8.
There exists ann0*∈ℕ such that for n≥n0*≥n0(4.20)γ(ℝN)≤sup1/p≤t1*,t2*≤pJ(t1*u0-t2*wn)<α(ℝN)+α∞(ℝN),
where u0 is a positive ground-state solution of (1.1) in ℝN.Proof.
By Lemma4.7, then
(4.21)J(t1*u0-t2*wn)=1p∥t1*u0-t2*wn∥1,pp-1qb(t1*u0-t2*wn)≤1p{∫ℝN(|∇(t1*u0)|p-2∇(t1*u0)-|∇(t2*wn)|p-2∇(t2*wn))(∇(t1*u0)-∇(t2*wn))}+1p{∫ℝN(|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*)qq∫ℝN(Q(z)-Q∞)w(z-zn)qdz-1qb(t1*u0-t2*wn)+1qb(t1*u0)+1qb(t2*wn).
Since supt≥0J(tu0)=α(ℝN) and supt≥0J∞(tw)=α∞(ℝN), using the inequality
(4.22)|c1-c2|q>c1q+c2q-K(c1q-1c2+c1c2q-1),
for any c1,c2>0, and some positive constant K, then
(4.23)sup1/p≤t1*,t2*≤pJ(t1*u0-t2*wn)≤α(ℝN)+α∞(ℝN)-1pqq∫ℝN(Q(z)-Q∞)w(z-zn)qdz+K′[∫ℝN(u0q-1wn+wnq-1u0)dz].(i)
SinceQ(z)≥Q∞+Cexp(-δ|z|) for some constants C>0 and 0<δ<θ, by Lemma 4.6, we have that there exists an n1≥n0 such that for n≥n1(4.24)∫ℝN(Q(z)-Q∞)w(z-zn)qdz≥C′exp(-min{δ,q(θ+ɛ)}|z¯|)≥C′exp(-δn).(ii)
Applying Lemma4.6, there exists an n2≥n1 such that for n≥n2(4.25)∫ℝNu0q-1wndz≤C1′∫ℝNexp(-(q-1)(θ-ɛ)|z|)exp(-(θ-ɛ)|z-zn|)dz≤C1′′exp(-(θ-ɛ)n).Similarly, we also obtain that there exists an n3≥n2 such that for n≥n3(4.26)∫ℝNwnq-1u0dz≤C1′′′exp(-(θ-ɛ)n).
By (i) and (ii), choosing 0<ɛ<θ-δ, we can find an n0*≥n3≥n0 such that for n≥n0*(4.27)sup1/p≤t1,t2≤pJ(t1*u0-t2*wn)<α(ℝN)+α∞(ℝN).Theorem 4.9.
Assume thatQ is a positive continuous function in ℝN and satisfies (Q1) and (Q2), then (1.1) has a positive solution and a nodal solution in ℝN.Proof.
By Lemmas4.2, 4.4, 4.5, and 4.8 and Theorem 3.4, we obtain the proof.



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