Multiple Solutions of Quasilinear Elliptic Equations in ℝ^𝑁

Abstract

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

1. Introduction

For 𝑁≥3, 2≤𝑝<𝑁, and 𝑝<𝑞<𝑝^∗=𝑁𝑝/(𝑁−𝑝), we consider the quasilinear elliptic equations

−Δ_𝑝𝑢+|𝑢|^𝑝−2𝑢=𝑄(𝑧)|𝑢|^𝑞−2𝑢inℝ^𝑁,𝑢∈𝑊^1,𝑝(ℝ^𝑁),−Δ_𝑝𝑢+|𝑢|^𝑝−2𝑢=𝑄_∞|𝑢|^𝑞−2𝑢inℝ^𝑁,𝑢∈𝑊^1,𝑝(ℝ^𝑁), (1.1)(1.2) where Δ_𝑝 is the 𝑝-Laplacian operator, that is,

Δ_𝑝𝑢=^𝑁_𝑖=1𝜕𝜕𝑧_𝑖(|∇𝑢|^𝑝−2𝜕𝑢𝜕𝑧_𝑖). (1.3) Let 𝑄 be a positive continuous function in ℝ^𝑁 and satisfy

𝑄(𝑧)≥𝑄_∞=lim_|𝑧|→∞𝑄(𝑧)>0,𝑄(𝑧)>𝑄_∞onasetofpositivemeasure. (Q1) Associated with (1.1) and (1.2), we define the functionals 𝑎,𝑏,𝑏^∞,𝐽, and 𝐽^∞, for 𝑢∈𝑊^1,𝑝(ℝ^𝑁),

𝑎(𝑢)=_ℝ𝑁(|∇𝑢|^𝑝+|𝑢|^𝑝)𝑑𝑧=‖𝑢‖^𝑝_1,𝑝,𝑏(𝑢)=_ℝ𝑁𝑄(𝑧)|𝑢|^𝑞𝑑𝑧,𝑏^∞(𝑢)=_ℝ𝑁𝑄_∞|𝑢|^𝑞1𝑑𝑧,𝐽(𝑢)=𝑝1𝑎(𝑢)−𝑞𝑏(𝑢),𝐽^∞1(𝑢)=𝑝1𝑎(𝑢)−𝑞𝑏^∞(𝑢). (1.4) 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 ℝ^𝑁. 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 ℝ^𝑁 (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. If 𝑄 is a positive continuous function in ℝ^𝑁 and satisfies (Q1), then we prove that (1.1) admits a positive ground-state solution in ℝ^𝑁. 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 in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽 as follows.

Definition 2.1. (i) For 𝛽∈ℝ, a sequence {𝑢_𝑛} is a (PS)_𝛽-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽 if 𝐽(𝑢_𝑛)=𝛽+𝑜_𝑛(1) and 𝐽^′(𝑢_𝑛)=𝑜_𝑛(1) strongly in 𝑊^−1,𝑝′(ℝ^𝑁) as 𝑛→∞, where 𝑊^−1,𝑝′(ℝ^𝑁) is the dual space of 𝑊^1,𝑝(ℝ^𝑁) and 1/𝑝+1/𝑝^′=1
(ii) 𝐽 satisfies the (PS)_𝛽-condition in 𝑊^1,𝑝(ℝ^𝑁) if every (PS)_𝛽-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽 contains a convergent subsequence.

Lemma 2.2. Let 𝛽∈ℝ and let {𝑢_𝑛} be a (PS)_𝛽-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽, then {𝑢_𝑛} is a bounded sequence in 𝑊^1,𝑝(ℝ^N). Moreover, 𝑎(𝑢_𝑛)=𝑏(𝑢_𝑛)+𝑜_𝑛(1)=(𝑞𝑝/(𝑞−𝑝))𝛽+𝑜_𝑛(1) as 𝑛→∞ and 𝛽≥0.

Proof. Since 𝑝≥2, we have that ^𝑝𝑎(𝑢_𝑛)≤1 if 𝑎(𝑢_𝑛)≤1 and ^𝑝𝑎(𝑢_𝑛)≤𝑎(𝑢_𝑛) if 𝑎(𝑢_𝑛)>1. For sufficiently large 𝑛, we get |𝛽|+2+𝑎(𝑢_𝑛)≥|𝛽|+1+^𝑝𝑎(𝑢_𝑛)≥𝐽(𝑢_𝑛1)−𝑞⟨𝐽^′(𝑢_𝑛),𝑢_𝑛1⟩=(𝑝−1𝑞)𝑎(𝑢_𝑛). (2.1) It follows that {𝑢_𝑛} is bounded in 𝑊^1,𝑝(ℝ^𝑁). Then ⟨𝐽^′(𝑢_𝑛),𝑢_𝑛⟩=𝑜_𝑛(1) as 𝑛→∞. Thus, 𝛽+𝑜_𝑛(1)=𝐽(𝑢_𝑛1)=(𝑝−1𝑞)𝑎(𝑢_𝑛)+𝑜_𝑛1(1)=(𝑝−1𝑞)𝑏(𝑢_𝑛)+𝑜_𝑛(1), (2.2) that is, 𝑎(𝑢_𝑛)=𝑏(𝑢_𝑛)+𝑜_𝑛(1)=(𝑞𝑝/(𝑞−𝑝))𝛽+𝑜_𝑛(1) as 𝑛→∞ and 𝛽≥0.

Define

𝛼(ℝ^𝑁)=inf_{𝑢∈𝐌(ℝ𝑁)}𝐽(𝑢), (2.3) where 𝐌(ℝ^𝑁)={𝑢∈𝑊^1,𝑝(ℝ^𝑁)⧵{0}∣𝑎(𝑢)=𝑏(𝑢)}, and

𝛼^∞(ℝ^𝑁)=inf_{𝑢∈𝐌∞(ℝ𝑁)}𝐽^∞(𝑢), (2.4) where 𝐌^∞(ℝ^𝑁)={𝑢∈𝑊^1,𝑝(ℝ^𝑁)⧵{0}∣𝑎(𝑢)=𝑏^∞(𝑢)}.

Lemma 2.3. Let 𝑢 be a sign-changing solution of (1.1). Then 𝐽(𝑢)≥2𝛼(ℝ^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 _ℝ𝑁(|∇𝑢|^𝑝−2∇𝑢∇𝑢⁻+|𝑢|^𝑝−2𝑢𝑢⁻)𝑑𝑧=_ℝ𝑁𝑄(𝑧)|𝑢|^𝑞−2𝑢𝑢⁻𝑑𝑧, (2.5) that is, 𝑢⁻∈𝐌(ℝ^𝑁) and 𝐽(𝑢⁻)≥𝛼(Ω). Similarly, 𝐽(𝑢⁺)≥𝛼(ℝ^𝑁). Hence, 𝐽(𝑢)=𝐽(𝑢⁺)+𝐽(𝑢⁻)≥2𝛼(ℝ^𝑁). (2.6)

Lemma 2.4. (i) For each 𝑢∈𝑊^1,𝑝(ℝ^𝑁)⧵{0}, there exists a positive number 𝑠_𝑢 such that 𝑠_𝑢𝑢∈𝐌(ℝ^𝑁) and sup_𝑠≥0𝐽(𝑠𝑢)=𝐽(𝑠_𝑢𝑢).
(ii) Let 𝛽>0 and let {𝑢_𝑛} be a sequence in 𝑊^1,𝑝(ℝ^𝑁)⧵{0} for 𝐽 such that 𝑎(𝑢_𝑛)=𝑏(𝑢_𝑛)+𝑜(1) and 𝐽(𝑢_𝑛)=𝛽+𝑜(1). Then there is a sequence {𝑠_𝑛} in ℝ⁺ such that 𝑠_𝑛=1+𝑜(1),{𝑠_𝑛𝑢_𝑛}⊂𝐌(ℝ^𝑁), and 𝐽(𝑠_𝑛𝑢_𝑛)=𝛽+𝑜(1) as 𝑛→∞.

Proof. (i) For each 𝑢∈𝑊₀^1,𝑝(ℝ^𝑁)⧵{0} and 𝑠≥0, let ℎ_𝑢𝑠(𝑠)=𝐽(𝑠𝑢)=^𝑝𝑝𝑠𝑎(𝑢)−^𝑞𝑞𝑏(𝑢). (2.7) Thus, ℎ^′_𝑢(𝑠)=𝑠^𝑝−1𝑎(𝑢)−𝑠^𝑞−1𝑏(𝑢). Define 𝑠_𝑢=(𝑎(𝑢)/𝑏(𝑢))^{1/(𝑞−𝑝)}>0, then ℎ^′_𝑢(𝑠_𝑢)=0, that is, 𝑠_𝑢𝑢∈𝐌(ℝ^𝑁).
(ii) By (i), there exists a sequence {𝑠_𝑛} in ℝ⁺ such that {𝑠_𝑛𝑢_𝑛}⊂𝐌(ℝ^𝑁), that is, 𝑠^𝑝_𝑛𝑎(𝑢_𝑛)=𝑠^𝑞_𝑛𝑏(𝑢_𝑛) for each 𝑛. Since 𝑎(𝑢_𝑛)=𝑏(𝑢_𝑛)+𝑜(1) and 𝐽(𝑢_𝑛)=𝛽+𝑜(1), we have that 𝑠_𝑛=1+𝑜(1). Hence, 𝐽(𝑠_𝑛𝑢_𝑛)=𝛽+𝑜(1) as 𝑛→∞.

Lemma 2.5. There exists 𝑐>0 such that ‖𝑢‖_1,𝑝≥𝑐>0 for each 𝑢∈𝐌(ℝ^𝑁), where 𝑐 is independent of 𝑢.

Proof. For each 𝑢∈𝐌(ℝ^𝑁), by the Sobolev inequality, we obtain that ‖𝑢‖^𝑝_1,𝑝=_ℝ𝑁𝑄(𝑧)|𝑢|^𝑞𝑑𝑧≤𝑐₁‖𝑢‖^𝑞_1,𝑝. (2.8) This implies that ‖𝑢‖_1,𝑝≥𝑐₁^{−1/(𝑞−𝑝)}=𝑐>0 for each 𝑢∈𝐌(ℝ^𝑁).

By Lemma 2.5, 𝛼(ℝ^𝑁)>0.

Lemma 2.6. Let 𝑢∈𝐌(ℝ^𝑁) such that 𝐽(𝑢)=min_{𝑣∈𝐌(ℝ𝑁)}𝐽(𝑣)=𝛼(ℝ^𝑁), (2.9) then 𝑢 is a nonzero solution of (1.1) in ℝ^𝑁.

Proof. Suppose that 𝜓(𝑣)=∫_ℝ𝑁(|∇𝑣|^𝑝+|𝑣|^𝑝)𝑑𝑧−∫_ℝ𝑁𝑄(𝑧)|𝑣|^𝑞𝑑𝑧, then ⟨𝜓^′(𝑣),𝑣⟩=(𝑝−𝑞)_ℝ𝑁(|∇𝑣|^𝑝+|𝑣|^𝑝)𝑑𝑧<0foreach𝑣∈𝐌(ℝ^𝑁). (2.10) Since 𝐽(𝑢)=min_{𝑣∈𝐌(ℝ}^𝑁₎𝐽(𝑣), by the Lagrange multiplier theorem, there is a 𝜆∈ℝ such that 𝐽^′(𝑢)=𝜆𝜓^′(𝑢) in 𝑊^−1,𝑝′(ℝ^𝑁). Then we have 0=⟨𝐽^′(𝑢),𝑢⟩=𝜆⟨𝜓^′(𝑢),𝑢⟩. (2.11) Thus, 𝜆=0 and 𝐽^′(𝑢)=0 in 𝑊^−1,𝑝′(ℝ^𝑁). Therefore, 𝑢 is a nonzero solution of (1.1) in ℝ^𝑁 with 𝐽(𝑢)=𝛼(ℝ^𝑁).

Lemma 2.7. There is a (PS)_𝛼(ℝ^𝑁₎-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽.

Proof. Let {𝑢_𝑛}⊂𝐌(ℝ^𝑁) be a minimizing sequence of 𝛼(ℝ^𝑁). Applying the Ekeland principle, there exists a sequence {𝑣_𝑛}⊂𝐌(ℝ^𝑁) such that ‖𝑣_𝑛−𝑢_𝑛‖_1,𝑝<1/𝑛, 𝐽(𝑣_𝑛)=𝛼(ℝ^𝑁)+𝑜(1), and 𝐽^′|_𝐌(ℝ^𝑁₎(𝑣_𝑛)=𝑜(1) strongly in 𝑊^−1,𝑝′(ℝ^𝑁) as 𝑛→∞. Let 𝜓(𝑢)=𝑎(𝑢)−𝑏(𝑢) for each 𝑢∈𝑊^1,𝑝(ℝ^𝑁)⧵{0}, then 𝐌(ℝ^𝑁)={𝑢∈𝑊^1,𝑝(ℝ^𝑁)⧵{0}∣𝜓(𝑢)=0}. (2.12) Thus, there exists a sequence {𝜃_𝑛}⊂ℝ such that 𝐽^′(𝑣_𝑛)=𝜃_𝑛𝜓^′(𝑣_𝑛)+𝑜_𝑛(1), where 𝑜_𝑛(1)→0 as 𝑛→∞. Since 𝑣_𝑛∈𝐌(ℝ^𝑁), we have that 0=⟨𝐽^′(𝑣_𝑛),𝑣_𝑛⟩=𝜃_𝑛⟨𝜓^′(𝑣_𝑛),𝑣_𝑛⟩+⟨𝑜_𝑛(1),𝑣_𝑛⟩,⟨𝜓^′(𝑣_𝑛),𝑣_𝑛⟩=(𝑝−𝑞)𝑎(𝑣_𝑛)≠0∀𝑛. (2.13) Hence, 𝜃_𝑛→0 as 𝑛→∞. This implies that 𝐽^′(𝑣_𝑛)=𝑜(1) strongly in 𝑊^−1,𝑝′(ℝ^𝑁) as 𝑛→∞, that is, {𝑣_𝑛}⊂𝐌(ℝ^𝑁) is a (PS)_𝛼(Ω)-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽.

Remark 2.8. The above definitions and lemmas also hold for 𝐽^∞,𝐌^∞(ℝ^𝑁), and 𝛼^∞(ℝ^𝑁).

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 for 𝐽). Assume that 𝑄 is a positive continuous function in ℝ^𝑁 and lim_|𝑧|→∞𝑄(𝑧)=𝑄_∞>0. Let {𝑢_𝑛} be a (PS)_𝛽-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽. Then there are a subsequence {𝑢_𝑛}, a positive integer 𝑙, sequences {𝑧^𝑖_𝑛}^∞_𝑛=1 in ℝ^𝑁, functions 𝑢 in 𝑊^1,𝑝(ℝ^𝑁), and 𝑤^𝑖≠0 in 𝑊^1,𝑝(ℝ^𝑁) for 1≤𝑖≤𝑙 such that |𝑧^𝑖_𝑛|→∞for1≤𝑖≤𝑙,−Δ_𝑝𝑢+|𝑢|^𝑝−2𝑢=𝑄(𝑧)|𝑢|^𝑞−2𝑢inℝ^𝑁,−Δ_𝑝𝑤^𝑖+|𝑤^𝑖|^𝑝−2𝑤^𝑖=𝑄_∞|𝑤^𝑖|^𝑞−2𝑤^𝑖inℝ^𝑁,𝑢_𝑛=𝑢+^𝑙_𝑖=1𝑤^𝑖(⋅−𝑧^𝑖_𝑛)+𝑜_𝑛(1)stronglyin𝑊^1,𝑝(ℝ^𝑁),𝐽(𝑢_𝑛)=𝐽(𝑢)+^𝑙_𝑖=1𝐽^∞(𝑤^𝑖)+𝑜_𝑛(1). (3.1) In addition, if 𝑢_𝑛≥0, then 𝑢≥0 and 𝑤^𝑖≥0 for 1≤𝑖≤𝑙.

Lemma 3.2. Let {𝑢_𝑛}⊂𝐌(ℝ^𝑁) be a (PS)_𝛽-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽 with 0<𝛽<𝛼^∞(ℝ^𝑁). Then there exist a subsequence {𝑢_𝑛} and a nonzero 𝑢∈𝑊^1,𝑝(ℝ^𝑁) such that 𝑢_𝑛→𝑢 strongly in 𝑊^1,𝑝(ℝ^𝑁) and 𝐽(𝑢)=𝛽, that is, 𝐽 satisfies the (PS)_𝛽-condition in 𝑊^1,𝑝(ℝ^𝑁).

Proof. Since {𝑢_𝑛}⊂𝐌(ℝ^𝑁) is a (PS)_𝛽-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽 with 0<𝛽<𝛼^∞(ℝ^𝑁), by Lemma 2.2, {𝑢_𝑛} is bounded in 𝑊^1,𝑝(ℝ^𝑁). Thus, there exist a subsequence {𝑢_𝑛} and 𝑢∈𝑊^1,𝑝(ℝ^𝑁) such that 𝑢_𝑛⇀𝑢 weakly in 𝑊^1,𝑝(ℝ^𝑁). It is easy to check that 𝑢 is a solution of (1.1) in ℝ^𝑁. Applying Palais-Smale Decomposition Lemma 3.1, we get 𝛼^∞>𝛽=𝐽(𝑢_𝑛)≥𝑙𝛼^∞. (3.2) Then 𝑙=0 and 𝑢≠0. Hence, 𝑢_𝑛→𝑢 strongly in 𝑊^1,𝑝(ℝ^𝑁) and 𝐽(𝑢)=𝛽.

Let 𝑤∈𝑊^1,𝑝(ℝ^𝑁) be the positive ground-state solution of (1.2) in ℝ^𝑁. 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^∞(ℝ^𝑁)∩𝐶^1,𝛾0_loc(ℝ^𝑁) for some 0<𝛾₀<1 and lim_|𝑧|→∞𝑤(𝑧)=0;(ii)for any 𝜀>0, there exist positive numbers 𝐶₁ and 𝐶₂ such that 𝐶₂exp(−(𝜃+𝜀)|𝑧|)≤𝑤(𝑧)≤𝐶₁exp(−(𝜃−𝜀)|𝑧|)∀𝑧∈ℝ^𝑁, (3.3) where 𝜃=(𝑝−1)^−1/𝑝.

Remark 3.3. Similarly, we also show that all positive solutions of (1.1) in ℝ^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 ℝ^𝑁 and satisfies (Q1). Then there exists a positive ground-state solution 𝑢₀ of (1.1) in ℝ^𝑁.

Proof. Let 𝑤∈𝑊^1,𝑝(ℝ^𝑁) be the positive ground-state solution of (1.2) in ℝ^𝑁, then 𝑤 is a minimizer of 𝛼^∞(ℝ^𝑁) and _ℝ𝑁(|∇𝑤|^𝑝+𝑤^𝑝)𝑑𝑧=_ℝ𝑁𝑄_∞𝑤^𝑞𝑑𝑧. (3.4) By Lemma 2.4(i), there exists a positive number 𝑠_𝑤 such that 𝑠_𝑤𝑤∈𝐌(ℝ^𝑁), that is, ∫_ℝ𝑁(|∇(𝑠_𝑤𝑤)|^𝑝+(𝑠_𝑤𝑤)^𝑝)𝑑𝑧=∫_ℝ𝑁𝑄(𝑧)(𝑠_𝑤𝑤)^𝑞𝑑𝑧. Since 𝑄(𝑧)>𝑄_∞ on a set of positive measure, we can deduce that 𝑠_𝑤<1. Therefore, 𝛼(ℝ^𝑁)≤𝐽(𝑠_𝑤1𝑤)=(𝑝−1𝑞)(𝑠_𝑤)^𝑝_ℝ𝑁(|∇𝑤|^𝑝+𝑤^𝑝1)𝑑𝑧<(𝑝−1𝑞)_ℝ𝑁(|∇𝑤|^𝑝+𝑤^𝑝1)𝑑𝑧=(𝑝−1𝑞)_ℝ𝑁𝑄_∞𝑤^𝑞𝑑𝑧=𝛼^∞(ℝ^𝑁). (3.5) Applying Lemma 3.2, there exists 𝑢₀∈𝑊^1,𝑝(ℝ^𝑁) such that 𝐽(𝑢₀)=𝛼(ℝ^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 ℝ^𝑁.

4. Existence of a Nodal Solution

In this section, assume that 𝑄 is a positive continuous function in ℝ^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|)∀𝑧∈ℝ^𝑁. (Q2) Let ℎ be a functional in 𝑊^1,𝑝(ℝ^𝑁) defined by

ℎ(𝑢)=𝑏(𝑢)𝑎(𝑢)for0𝑢≠0,for𝑢=0. (4.1) We define

𝐌₀={𝑢∈𝑊^1,𝑝(ℝ^𝑁)∣ℎ(𝑢⁺)=1,ℎ(𝑢⁻)=1}⊂𝐌(ℝ^𝑁),𝒩={𝑢∈𝑊^1,𝑝(ℝ^𝑁)∣|ℎ(𝑢^±1)−1|<2}⊃𝐌₀, (4.2) where 𝑢⁺=max{𝑢,0} and 𝑢⁻=max{−𝑢,0}.

Lemma 4.1. (i) If 𝑢∈𝑊^1,𝑝(ℝ^𝑁) changes sign, then there exist positive numbers 𝑠^±(𝑢)=𝑠^± such that 𝑠⁺𝑢⁺∈𝐌(ℝ^𝑁) and 𝑠⁻𝑢⁻∈𝐌(ℝ^𝑁).
(ii) There exists 𝑐^′>0 such that ‖𝑢^±‖_1,𝑝≥𝑐^′>0 for each 𝑢∈𝒩.

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 𝑢∈𝒩, by Lemma 2.4(i), there exists 𝑠^±(𝑢)=𝑠^±>0 such that 𝑠^±𝑢^±∈𝐌(ℝ^𝑁). Then 12<(𝑠^±)^𝑝−𝑞=𝑏(𝑢^±)𝑎(𝑢^±)<32foreach𝑢∈𝒩. (4.3) By Lemma 2.5, we have ‖𝑠^±𝑢^±‖_1,𝑝≥𝑐forsome𝑐>0andeach𝑢∈𝒩. (4.4) Hence, ‖𝑢^±‖_1,𝑝≥𝑐/𝑠^±≥𝑐^′>0 for each 𝑢∈𝒩.

Consider these minimization problem

𝛾(ℝ^𝑁)=inf_𝑢∈𝐌0𝐽(𝑢). (4.5) By Lemma 4.1, 𝛾(ℝ^𝑁)>0.

Lemma 4.2. There exists a sequence {𝑢_𝑛}⊂𝒩 such that 𝐽(𝑢_𝑛)=𝛾(ℝ^𝑁)+𝑜_𝑛(1) and 𝐽^′(𝑢_𝑛)=𝑜_𝑛(1) strongly in 𝑊^−1,𝑝(ℝ^𝑁) as 𝑛→∞.

Proof. It is similar to the proof of Zhu [13].

Lemma 4.3. Let 𝑓 and 𝑔 be real-valued functions in ℝ^𝑁. If 𝑔(𝑧)>0 in ℝ^𝑁, 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 {𝑢_𝑛}⊂𝒩 be a (PS)_𝛾(ℝ^𝑁₎-sequence in 𝑊^1𝑝(ℝ^𝑁) for 𝐽 satisfying 𝛼(ℝ^𝑁)<𝛾(ℝ^𝑁)<𝛼(ℝ^𝑁)+𝛼^∞(ℝ^𝑁)(<2𝛼^∞(ℝ^𝑁)). (4.6) Then there exists 𝑢^∗∈𝐌₀ such that 𝑢_𝑛 converges to 𝑢^∗ strongly in 𝑊^1,𝑝(ℝ^𝑁) and 𝑢^∗ is a higher-energy solution of (1.1) such that 𝐽(𝑢^∗)=𝛾(ℝ^𝑁).

Proof. By the definition of the (PS)_𝛾(ℝ^𝑁₎-sequence in 𝑊^1,𝑝(ℝ^𝑁) for 𝐽, it is easy to see that {𝑢_𝑛} is a bounded sequence in 𝑊^1,𝑝(ℝ^𝑁) and satisfies _ℝ𝑁(|∇𝑢^±_𝑛|^𝑝+|𝑢^±_𝑛|^𝑝)𝑑𝑧=_ℝ𝑁𝑄(𝑧)|𝑢^±_𝑛|^𝑞𝑑𝑧+𝑜_𝑛(1). (4.7) By Lemma 4.1(ii), there exists 𝑐^′>0 such that 𝑐^′≤_ℝ𝑁(|∇𝑢^±_𝑛|^𝑝+|𝑢^±_𝑛|^𝑝)𝑑𝑧=_ℝ𝑁𝑄(𝑧)|𝑢^±_𝑛|^𝑞𝑑𝑧+𝑜_𝑛(1). (4.8) Using the Palais-Smale Decomposition Lemma 3.1, then we have 𝛾(ℝ^𝑁)=𝐽(𝑢^∗)+∑^𝑙_𝑖=1𝐽^∞(𝑤_𝑖), where 𝑢^∗ is a solution of (1.1) in ℝ^𝑁 and 𝑤_𝑖 is a solution of (1.2) in ℝ^𝑁. Since 𝐽^∞(𝑤_𝑖)≥𝛼^∞(ℝ^𝑁) for each 𝑖∈ℕ and 𝛼(ℝ^𝑁)<𝛼^∞(ℝ^𝑁), 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(ℝ^𝑁)≥2𝛼^∞(ℝ^𝑁), 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 {𝑧¹_𝑛} in ℝ^𝑁 such that |𝑧¹_𝑛|→∞, and ‖𝑢_𝑛−𝑢^∗−𝑤₁(⋅−𝑧¹_𝑛)‖_1,𝑝=𝑜_𝑛(1). (4.9)By the Sobolev continuous embedding inequality, we obtain ‖𝑢_𝑛−𝑢^∗−𝑤₁(⋅−𝑧¹_𝑛)‖_𝐿𝑞=𝑜_𝑛(1). (4.10) Since 𝑤₁>0, by Lemma 4.3, then ‖(𝑢_𝑛−𝑢^∗)⁻‖_𝐿𝑞=𝑜_𝑛(1)as𝑛→∞. (4.11)(a)Suppose that 𝑢^∗≡0; we obtain ‖𝑢⁻_𝑛‖_𝐿𝑞=𝑜_𝑛(1) as 𝑛→∞. Then 0<𝑐^′≤_ℝ𝑁𝑄(𝑧)|𝑢⁻_𝑛|^𝑞𝑑𝑧=𝑜_𝑛(1), (4.12) which is a contradiction.(b)Suppose that 𝑢^∗≢0. We have 𝛾(ℝ^𝑁)=𝐽(𝑢^∗)+𝐽^∞(𝑤₁)≥𝛼(ℝ^𝑁)+𝛼^∞(ℝ^𝑁), which is a contradiction.
By (i) and (ii), then 𝑙=0. Thus, ‖𝑢_𝑛−𝑢^∗‖_1,𝑝=𝑜_𝑛(1) as 𝑛→∞ and 𝐽(𝑢^∗)=𝛾(ℝ^𝑁). Finally, we claim that 𝑢^∗ is a sign-changing solution of (1.1) in ℝ^𝑁. If 𝑢^∗>0(or<0), by Lemma 4.3, then ‖𝑢⁻_𝑛‖_𝐿𝑞=𝑜_𝑛(1)(or‖𝑢⁻_𝑛‖_𝐿𝑞=𝑜_𝑛(1)). Similarly, we have the inequality (4.12), which is a contradiction. Moreover, by Lemma 2.3, 2𝛼(ℝ^𝑁)≤𝛾(ℝ^𝑁).

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

𝐶₂exp(−(𝜃+𝜀)|𝑧|)≤𝑤(𝑧)≤𝐶₁exp(−(𝜃−𝜀)|𝑧|)∀𝑧∈ℝ^𝑁, (4.13) where 𝜃=(𝑝−1)^−1/𝑝. Define

𝑤_𝑛(𝑧)=𝑤(𝑧−𝑧_𝑛)where𝑧_𝑛=(0,…,0,𝑛)∈ℝ^𝑁. (4.14) Clearly, 𝑤_𝑛(𝑧)∈𝑊^1,𝑝(ℝ^𝑁).

Lemma 4.5. There are 𝑛₀∈ℕ and real numbers 𝑡^∗₁ and 𝑡^∗₂ such that for 𝑛≥𝑛₀𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛∈𝐌₀,𝛾(ℝ^𝑁)≤𝐽(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛), (4.15) where 1/𝑝≤𝑡^∗₁, 𝑡^∗₂≤𝑝, and 𝑢₀ is the positive ground-state solution of (1.1) in ℝ^𝑁.

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 that sup_{1/𝑝≤𝑡∗1,𝑡∗2≤𝑝}𝐽(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)<𝛼(ℝ^𝑁)+𝛼^∞(ℝ^𝑁) for sufficiently large 𝑛.

Lemma 4.6. Let 𝐸 be a domain in ℝ^𝑁. If 𝑓∶𝐸→ℝ satisfies _𝐸|𝑓(𝑧)𝑒^𝜎|𝑧||𝑑𝑧<∞forsome𝜎>0, (4.16) then (_𝐸𝑓(𝑧)𝑒^{−𝜎|𝑧−𝑧|}𝑑𝑧)𝑒^𝜎|𝑧|=_𝐸𝑓(𝑧)𝑒^{𝜎⟨𝑧,𝑧⟩/|𝑧|}𝑑𝑧+𝑜(1)as|𝑧|→∞. (4.17)

Proof. Since 𝜎|𝑧|≤𝜎|𝑧|+𝜎|𝑧−𝑧|, we have |𝑓(𝑧)𝑒^{−𝜎|𝑧−𝑧|}𝑒^𝜎|𝑧||≤|𝑓(𝑧)𝑒^𝜎|𝑧||. (4.18) 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,𝑦∈ℝ^𝑁, one has the following inequality: |𝑥−𝑦|^𝜌≤(|𝑥|^𝜌−2𝑥−|𝑦|^𝜌−2𝑦)(𝑥−𝑦),where𝜌≥2. (4.19)

Proof. See Yang [16, Lemma 4.2.].

Lemma 4.8. There exists an 𝑛^∗₀∈ℕ such that for 𝑛≥𝑛^∗₀≥𝑛₀𝛾(ℝ^𝑁)≤sup_{1/𝑝≤𝑡∗1,𝑡∗2≤𝑝}𝐽(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)<𝛼(ℝ^𝑁)+𝛼^∞(ℝ^𝑁), (4.20) where 𝑢₀ is a positive ground-state solution of (1.1) in ℝ^𝑁.

Proof. By Lemma 4.7, then 𝐽(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)=1𝑝‖𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛‖^𝑝_1,𝑝−1𝑞𝑏(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)≤1𝑝{_ℝ𝑁(|∇(𝑡^∗₁𝑢₀)|^𝑝−2∇(𝑡^∗₁𝑢₀)−|∇(𝑡^∗₂𝑤_𝑛)|^𝑝−2∇(𝑡^∗₂𝑤_𝑛))(∇(𝑡^∗₁𝑢₀)−∇(𝑡^∗₂𝑤_𝑛+1))}𝑝{_ℝ𝑁(|𝑡^∗₁𝑢₀|^𝑝−2(𝑡^∗₁𝑢₀)−|𝑡^∗₂𝑤_𝑛|^𝑝−2(𝑡^∗₂𝑤_𝑛))(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛1)}−𝑞𝑏(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)≤𝐽(𝑡^∗₁𝑢₀)+𝐽^∞(𝑡^∗₂𝑤)−(𝑡^∗₂)^𝑞𝑞_ℝ^𝑁(𝑄(𝑧)−𝑄_∞)𝑤(𝑧−𝑧_𝑛)^𝑞−1𝑑𝑧𝑞𝑏(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛1)+𝑞𝑏(𝑡^∗₁𝑢₀1)+𝑞𝑏(𝑡^∗₂𝑤_𝑛). (4.21) Since sup_𝑡≥0𝐽(𝑡𝑢₀)=𝛼(ℝ^𝑁) and sup_𝑡≥0𝐽^∞(𝑡𝑤)=𝛼^∞(ℝ^𝑁), using the inequality |𝑐₁−𝑐₂|^𝑞>𝑐^𝑞₁+𝑐^𝑞₂−𝐾(𝑐₁^𝑞−1𝑐₂+𝑐₁𝑐₂^𝑞−1), (4.22) for any 𝑐₁,𝑐₂>0, and some positive constant 𝐾, then sup_{1/𝑝≤𝑡∗1,𝑡∗2≤𝑝}𝐽(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)≤𝛼(ℝ^𝑁)+𝛼^∞(ℝ^𝑁1)−𝑝^𝑞𝑞_ℝ^𝑁(𝑄(𝑧)−𝑄_∞)𝑤(𝑧−𝑧_𝑛)^𝑞𝑑𝑧+𝐾^′[_ℝ𝑁(𝑢₀^𝑞−1𝑤_𝑛+𝑤_𝑛^𝑞−1𝑢₀)𝑑𝑧]. (4.23)(i)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 𝑛₁≥𝑛₀ such that for 𝑛≥𝑛_1ℝ𝑁(𝑄(𝑧)−𝑄_∞)𝑤(𝑧−𝑧_𝑛)^𝑞𝑑𝑧≥𝐶^′exp(−min{𝛿,𝑞(𝜃+𝜀)}|𝑧|)≥𝐶^′exp(−𝛿𝑛). (4.24)(ii)Applying Lemma 4.6, there exists an 𝑛₂≥𝑛₁ such that for 𝑛≥𝑛_2ℝ𝑁𝑢₀^𝑞−1𝑤_𝑛𝑑𝑧≤𝐶^′_1ℝ𝑁exp(−(𝑞−1)(𝜃−𝜀)|𝑧|)exp(−(𝜃−𝜀)|𝑧−𝑧_𝑛|)𝑑𝑧≤𝐶₁^′′exp(−(𝜃−𝜀)𝑛). (4.25)Similarly, we also obtain that there exists an 𝑛₃≥𝑛₂ such that for 𝑛≥𝑛_3ℝ𝑁𝑤_𝑛^𝑞−1𝑢₀d𝑧≤𝐶₁^′′′exp(−(𝜃−𝜀)𝑛). (4.26) By (i) and (ii), choosing 0<𝜀<𝜃−𝛿, we can find an 𝑛^∗₀≥𝑛₃≥𝑛₀ such that for 𝑛≥𝑛^∗₀sup_{1/𝑝≤𝑡1,𝑡2≤𝑝}𝐽(𝑡^∗₁𝑢₀−𝑡^∗₂𝑤_𝑛)<𝛼(ℝ^𝑁)+𝛼^∞(ℝ^𝑁). (4.27)

Theorem 4.9. Assume that 𝑄 is a positive continuous function in ℝ^𝑁 and satisfies (Q1) and (Q2), then (1.1) has a positive solution and a nodal solution in ℝ^𝑁.

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