Ground-State Solutions for a Class of N-Laplacian Equation with Critical Growth

Abstract

We investigate the existence of ground-state solutions for a class of N-Laplacian equation with critical growth in ℝ^𝑁. Our proof is based on a suitable Trudinger-Moser inequality, Pohozaev-Pucci-Serrin identity manifold, and mountain pass lemma.

1. Introduction

Consider the following 𝑁-Laplacian equation: −𝑢Δ>_𝑁0𝑢,+i|n𝑢|ℝ^𝑁−2_𝑁𝑢=𝑓(𝑢),inℝ^𝑁,,𝑢∈𝑊^1,𝑁ℝ^𝑁,(1.1) where 𝑁≥2. Δ_𝑁𝑢=div(|∇𝑢|^𝑁−2∇𝑢) is the 𝑁-Laplacian, the nonlinear term 𝑓(𝑢) has critical growth.

The interest in these problems lies in that fact that the order of the Laplacian is the same as the dimension 𝑁 of the underlying space. The classical case of this problem that 𝑁=2, and the the problem (1.1) reduces to −Δ𝑢+𝑢=𝑓(𝑢),inℝ²,(1.2) has been treated by Atkinson and Peletier [1] and Berestycki and Lions [2]. They obtained the existence of ground-state solution which the nonlinear term 𝑓(𝑢) is subcritical growth. Alves et al. [3] extend their results to the critical growth. As 𝑁≠2, do Ó and Medeiros [4] consider the following 𝑁-Laplacian equation problem: −Δ_𝑁𝑢=𝑔(𝑢),inℝ^𝑁,(1.3) where 𝑔∶ℝ↦ℝ has a subcritical growth and obtain a mountain pass characterization of the ground-state solution for the problem (1.3). In the present paper, we will improve and complement some of the results cited above.

Assume the function 𝑓∶ℝ↦ℝ is continuous and satisfies the following conditions: (𝑔₁)lim_𝑠→0⁺(𝑓(𝑠)/𝑠|𝑠|^𝑁−2)=0; (𝑔₂) There exist constants 𝛼₀,𝑏₁,𝑏₂>0 such that |𝑓(𝑠)|≤𝑏₁|𝑠|^𝑁−1+𝑏₂[exp(𝛼₀|𝑠|^{𝑁/(𝑁−1)})−𝑆_𝑁−2(𝛼₀,𝑠)], where 𝑆_𝑁−2(𝛼₀,𝑠)=∑^{𝑁−2𝑘=0}(𝛼^𝑘0/𝑘!)|𝑠|^{𝑁𝑘/(𝑁−1)};(𝑔₃)There exist 𝜆>0 and 𝑞>𝑁 such that 𝑓(𝑠)≥𝜆𝑠^𝑞−1, for every 𝑠≥0.

Remark 1.1. Condition (𝑔₂) implies that 𝑓 has a critical growth with critical exponent 𝛼₀.
Consider the energy functional 𝐼∶𝑊^1,𝑁(ℝ^𝑁)↦ℝ𝐼(𝑢)=1𝑁_ℝ^𝑁||∇𝑢||^𝑁+|𝑢|^𝑁𝑑𝑥−_ℝ𝑁𝐹(𝑢)𝑑𝑥,(1.4) where 𝐹(𝑠)=∫^𝑠0𝑓(𝑡)𝑑𝑡. By a ground-state solution, we mean a solution such 𝜔∈𝑊^1,𝑁(ℝ^𝑁) such that 𝐼(𝜔)≤𝐼(𝑢) for every nontrivial solution 𝑢 of the problem (1.1). Let 𝐶_𝑞>0 denote the best constant of Sobolev embeddings: 𝑊^1,𝑁ℝ^𝑁↪𝐿^𝑞ℝ^𝑁,(1.5) for 𝑞∈(𝑁,+∞), that is, 𝐶_𝑞_ℝ𝑁|𝑢|^𝑞𝑑𝑥^𝑁/𝑞≤_ℝ𝑁||∇𝑢||^𝑁+|𝑢|^𝑁𝑑𝑥,(1.6) for all 𝑢∈𝑊^1,𝑁(ℝ^𝑁).

Now we state our main theorem in this paper.

Theorem 1.2. If 𝑓 satisfies (𝑔₁),(𝑔₂), and (𝑔₃), with 𝜆>𝑞−𝑁𝑞^{(𝑞−𝑁)/𝑁}𝐶^{𝑞𝑞/𝑁},(1.7) then the problem (1.1) possesses a nontrivial ground-state solution.

In this paper, we complement some results [4] from subcritical case to the critical case. Furthermore, the ground-state solution to the problem (1.1) is obtained without assuming that the function 𝑠↦𝑓(𝑠)|𝑠|^𝑁−1(1.8) is increasing for 𝑠>0 (see [5]), and the so-called Ambrosetti-Rabinowitz condition: there exists 𝜃>𝑁, such that for all 𝑥∈ℝ^𝑁, 0<𝜃𝐹(𝑥,𝑢)≤𝑢𝑓(𝑥,𝑢).(1.9)

The paper is organized as follows. Section 2 contains some technical results which allows us to give a variational approach for our results. In Section 3, we prove our main results.

2. The Variational Framework

For 1≤𝑝≤∞, 𝐿^𝑝(ℝ^𝑁) denotes the Lebesgue spaces with the norm ‖𝑢‖_𝐿^𝑝_(ℝ^𝑁₎=(∫_ℝ𝑁|𝑢|^𝑝𝑑𝑥)^1/𝑝, 𝑊^1,𝑝(ℝ^𝑁) denotes the Sobolev spaces with the norm ‖𝑢‖_𝑊^1,𝑝_(ℝ^𝑁₎=(∫_ℝ𝑁(|∇𝑢|^𝑝+|𝑢|^𝑝)𝑑𝑥)^1/𝑝. As 𝑝=𝑁, we have the following version of Trudinger-Moser inequality.

Lemma 2.1 (see [6]). If 𝑁≥2, 𝛼>0 and 𝑢∈𝑊^1,𝑁(ℝ^𝑁), then _ℝ𝑁exp𝛼|𝑢|^{𝑁/(𝑁−1)}−𝑆_𝑁−2(𝛼,𝑢)𝑑𝑥<∞.(2.1) Moreover, if ‖∇𝑢‖^𝑁𝐿_{𝑁(ℝ𝑁)}≤1,‖𝑢‖_𝐿^𝑁_(ℝ^𝑁₎≤𝑀<∞ and 𝛼<𝛼_𝑁, then there exists a constant 𝐶, which depends only on 𝑁, 𝑀, and 𝛼, such that _ℝ𝑁exp𝛼|𝑢|^{𝑁/(𝑁−1)}−𝑆_𝑁−2(𝛼,𝑢)𝑑𝑥≤𝐶(𝑁,𝑀,𝛼),(2.2) where 𝛼_𝑁=𝑁𝜔^{1/(𝑁−1)𝑁−1} and 𝜔^1/(𝑁−1) is the measure of the unit sphere in ℝ^𝑁.

In the sequel, since we seek positive solutions, and assume that 𝑓(𝑠)=0 for 𝑠≤0. Consider the following minimization problem: min1𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥∶_ℝ𝑁𝐺(𝑢)𝑑𝑥=0,(2.3) where 𝑔(𝑠)=𝑓(𝑠)−𝑠|𝑠|^𝑁−2, 𝐺(𝑠)=∫^𝑠0𝑔(𝑡)𝑑𝑡=𝐹(𝑠)−(1/𝑁)𝑠^𝑁. Since the problem (1.1) is an autonomous problem, under the Schwarz symmetric process, we can minimize the problem (2.3) on the space 𝑊^1,𝑁_rad(ℝ^𝑁), the subspace of 𝑊^1,𝑁(ℝ^𝑁) formed by radially symmetric functions. Indeed, let 𝑢^∗ be the Schwarz symmetrization of 𝑢, we have _ℝ𝑁𝐺𝑢^∗𝑑𝑥=_ℝ𝑁𝐺(𝑢)𝑑𝑥,_ℝ𝑁||∇𝑢^∗||^𝑁𝑑𝑥≤_ℝ𝑁||∇𝑢||^𝑁𝑑𝑥.(2.4) Hence, we can minimize the problem (2.3) on the space 𝑊^1,𝑁_rad(ℝ^𝑁) (see [7]). Now, we defined the following notations 𝑚=inf𝐴{=𝐼i(n𝑢f)∶𝑢1isnontrivialsolutionoftheproblem(1.1)}𝑁_ℝ^𝑁𝑏=in||f∇_𝛾∈Γ𝑢||^𝑁𝑑𝑥∶_ℝ𝑁𝐺(𝑢)𝑑𝑥=0max_[]𝑡∈0,1𝐼(𝛾(𝑡)),(2.5) where Γ={𝛾∈𝐶([0,1],𝑊^1,𝑁_rad(ℝ^𝑁))∶𝛾(0)=0,𝐼(𝛾(1))<0}.

We recall that Pohozaev-Pucci-Serrin identity shows that any solutions 𝑢 of the problem (1.1) should satisfies the Pohozaev-Pucci-Serrin identity: (𝑁−𝑝)_ℝ𝑁||∇𝑢||^𝑝𝑑𝑥=𝑁𝑝_ℝ𝑁𝐺(𝑢)𝑑𝑥.(2.6)

Then, as 𝑝=𝑁, we have ∫_ℝ𝑁𝐺(𝑢)𝑑𝑥=0.

Hence, we have the Pohozaev identity manifold: 𝒫==𝑢𝑢∈∈𝑊𝑊^1,𝑁_1,𝑁ℝ^𝑁⧵{0}∶(𝑁−𝑝)_ℝ𝑁||∇𝑢||^𝑝𝑑𝑥=𝑁𝑝_ℝ𝑁𝐺(𝑢)𝑑𝑥ℝ^𝑁⧵{0}∶_ℝ𝑁𝐺(𝑢)𝑑𝑥=0.(2.7)

So, we have 𝐴=inf_𝑢∈𝒫1𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥,𝑚=inf_𝑢∈𝜏𝐼(𝑢),(2.8) where 𝜏={𝑢∈𝑊^1,𝑁(ℝ^𝑁)⧵{0}∶𝐼′(𝑢)=0}.

In what follows, we will show that 𝐴 is attained, and afterwards we prove that 𝑚=𝐴=𝑏,(2.9) thereby proving that the problem (1.1) has a ground-state solution.

3. The Proof of Theorem 1.2

In this section, we prove that 𝐴 is attained, the equality (2.9) is satisfied. Hence the proof of Theorem 1.2 is obtained.

In the following, we consider the following minimax value: 𝑐=inf_{0≠𝑣∈𝑊1,𝑁ℝ𝑁}max_𝑡≥0𝐼(𝑡𝑣).(3.1)

Now, we show a sufficient condition, on a sequence {𝑣_𝑛} to get a convergence like 𝐹(𝑣_𝑛)→𝐹(𝑣) in 𝐿¹(ℝ^𝑁).

Lemma 3.1. Assume that 𝑓 satisfies (𝑔₁) and(𝑔₂), and let {𝑣_𝑛} be a sequence in 𝑊^1,𝑁_rad(ℝ^𝑁) such that ‖∇𝑣_𝑛‖^𝑁𝐿_{𝑁(ℝ𝑁)}≤1,‖𝑣_𝑛‖_𝐿^𝑁_(ℝ^𝑁₎≤𝑀<∞, then we have _ℝ𝑁𝐹𝑣_𝑛𝑑𝑥⟶_ℝ𝑁𝐹(𝑣)𝑑𝑥,(3.2) where 𝑣_𝑛⇀𝑣 in 𝑊^1,𝑁(ℝ^𝑁).

Proof. Without loss of generality, we assume that there exist 𝑣∈𝑊^1,𝑁_rad(ℝ^𝑁) such that 𝑣_𝑛𝑣⇀_𝑛𝑣,in𝑊^1,𝑁_radℝ^𝑁,⇀𝑣,a.e.inℝ^N.(3.3) Let 𝑣^∗ is the Schwarz symmetrization of 𝑣, then we have _ℝ𝑁𝑣exp𝛼₀|𝑣|^{𝑁/(𝑁−1)}−𝑆_𝑁−2𝛼₀,𝑣𝑑𝑥=_ℝ𝑁𝑣^∗exp𝛼⁰||𝑣^∗||^{𝑁/(𝑁−1)}−𝑆^𝑁−2𝛼₀,𝑣^∗𝑑𝑥,_ℝ𝑁|𝑣|^𝑁𝑑𝑥=_ℝ𝑁||𝑣^∗||^𝑁𝑑𝑥.(3.4)
From (𝑔₁), we obtain that for 𝜖>0, there exists 𝛿>0, such that 𝑓(𝑠)≤𝜖|𝑠|^𝑁−1,for|𝑠|<𝛿,(3.5) so, we have 𝐹(𝑠)≤𝜖𝑁|𝑠|^𝑁,for|𝑠|<𝛿.(3.6)
From (𝑔₂), we obtain 𝐹(𝑠)≤𝐶₁|𝑠|^𝑁+𝐶₂|𝑠|exp𝛼₀|𝑠|^{𝑁/(𝑁−1)}−𝑆_𝑁−2𝛼₀,𝑢,for|𝑠|≥𝛿.(3.7) There two estimates yield 𝐹(𝑠)≤𝐶₃|𝑠|^𝑁+𝐶₂|𝑠|exp𝛼₀|𝑠|^{𝑁/(𝑁−1)}−𝑆_𝑁−2𝛼₀,𝑢,for𝑠>0.(3.8)
On one hand, from Lemma 2.1, we obtain that there exists a constant 𝐶, which depends only on 𝑁, 𝑀, and 𝛼 such that _ℝ𝑁exp𝛼||𝑣_𝑛||^{𝑁/(𝑁−1)}≤𝐶.(3.9) When ‖∇𝑣_𝑛‖^𝑁𝐿_{𝑁(ℝ𝑁)}≤1,‖𝑣_𝑛‖_𝐿^𝑁_(ℝ^𝑁₎≤𝑀<∞ and 𝛼<𝛼_𝑁. Hence, we have _{|𝑥|≤𝑟}𝐹𝑣_𝑛≤𝐶₃_ℝ𝑁||𝑣^∗𝑛||^𝑁𝑑𝑥+𝐶₂_{|𝑥|≤𝑟}||𝑣^∗𝑛||exp𝛼₀||𝑣^∗𝑛||^{𝑁/(𝑁−1)}≤𝐶₃−𝑆^𝑁−2𝛼₀,𝑣^∗𝑛𝑑𝑥𝑀^𝑁+𝐶₂_{|𝑥|≤𝑟}||𝑣^∗𝑛||exp𝛼₀||𝑣^∗𝑛||^{𝑁/(𝑁−1)}≤𝐶₃𝑑𝑥𝑀^𝑁+𝐶₃_{|𝑥|≤𝑟}||𝑣^∗𝑛||^𝜇^1/𝜇_{|𝑥|≤𝑟}exp𝛽𝛼₀||𝑣^∗𝑛||^{𝑁/(𝑁−1)}𝑑𝑥^1/𝛽≤𝐶₃𝑀^𝑁+𝐶₄𝑀_{|𝑥|≤𝑟}exp𝛽𝛼₀||𝑣^∗𝑛||^{𝑁/(𝑁−1)}𝑑𝑥^1/𝛽≤𝐶₅,(3.10) where 𝐶_𝑖(𝑖=2,3,4,5) are positive constants, the continuous imbedding 𝑊^1,𝑁(ℝ^𝑁)↪𝐿^𝜇(ℝ^𝑁), 1/𝜇+1/𝛽=1 and 𝛽𝛼₀<𝛼_𝑁.
Then, by Dominated convergence theorem, we obtain _{|𝑥|≤𝑟}𝐹𝑣_𝑛𝑑𝑥⟶_{|𝑥|≤𝑟}𝐹(𝑣)𝑑𝑥.(3.11)
On the other hand, _|𝑥|>𝑟𝐹𝑣_𝑛𝑑𝑥≤𝐶₃_|𝑥|>𝑟||𝑣_𝑛||^𝑁𝑑𝑥+𝐶₂_|𝑥|>𝑟||𝑣_𝑛||exp𝛼₀||𝑣_𝑛||^{𝑁/(𝑁−1)}−𝑆^𝑁−2𝛼₀,𝑣_𝑛𝑑𝑥_|𝑥|>𝑟||𝑣_𝑛||exp𝛼₀||𝑣_𝑛||^{𝑁/(𝑁−1)}=−𝑆^𝑁−2𝛼₀,𝑣_𝑛𝑑𝑥_∞𝑗=_𝑁−1𝛼^𝑗0𝑗!_|𝑥|>𝑟=||𝑣_𝑛||⋅||𝑣_𝑛||^{𝑁𝑗/(𝑁−1)}𝑑𝑥_∞𝑗=_𝑁−1𝛼^𝑗0𝑗!_|𝑥|>𝑟||𝑣^∗𝑛||||𝑣^∗𝑛||^{𝑁𝑗/(𝑁−1)}𝑑𝑥,(3.12) where 𝑣^∗𝑛 is the Schwarz symmetrization of 𝑣_𝑛. Notice that the estimate _|𝑥|>𝑟1|𝑥|^{1+𝑁𝑗/(𝑁−1)}𝑑𝑥=𝜔_𝑁−1^∞𝑟𝑡^𝑁−1𝑡^{1+𝑁𝑗/(𝑁−1)}=𝜔𝑑𝑡_𝑁−1𝑁𝑗/(𝑁−1)−𝑁+1𝑟^{𝑁−1−𝑁𝑗/(𝑁−1)}≤𝜔^𝑁−1𝑟,(3.13) for all 𝑗≥𝑁−1, together with the Radial Lemma [4] leads to ^∞_𝑗_=𝑁−1𝛼^𝑗0𝑗!_|𝑥|>𝑟≤𝑀𝑁||𝑣^∗𝑛||||𝑣^∗𝑛||^{𝑁𝑗/(𝑁−1)}𝑑𝑥𝜔_𝑁−1^∞1/𝑁_{𝑗=𝑁−1}𝛼^𝑗0𝑗!𝜔𝑁_𝑁−1^{𝑗/(𝑁−1)}𝑀^{𝑁𝑗/(𝑁−1)}_|𝑥|>𝑟≤𝐶(𝑁)|𝑥|^{−1−𝑁𝑗/(𝑁−1)}𝑑𝑥𝑟.(3.14) Thus, given 𝛿>0, there exists 𝑟>0 such that _|𝑥|>𝑟||𝑣_𝑛||^𝑁𝑑𝑥<𝛿,_|𝑥|>𝑟exp𝛼⁰||𝑣_𝑛||^{𝑁/(𝑁−1)}−𝑆^𝑁−2𝛼₀,𝑣_𝑛𝑑𝑥<𝛿.(3.15) Which implies that _|𝑥|>𝑟𝐹𝑣_𝑛𝑑𝑥≤𝐶𝛿,_|𝑥|>𝑟𝐹(𝑣)𝑑𝑥≤𝐶𝛿.(3.16)
Using the estimate ||||_ℝ𝑁𝐹𝑣_𝑛−𝐹(𝑣)𝑑𝑥||||≤||||_{|𝑥|≤𝑟}𝐹𝑣_𝑛−𝐹(𝑣)𝑑𝑥||||+||||_|𝑥|>𝑟𝐹𝑣_𝑛−𝐹(𝑣)𝑑𝑥||||,(3.17) we get lim_𝑛→∞||||_ℝ𝑁𝐹𝑣_𝑛−_ℝ𝑁𝐹(𝑣)𝑑𝑥||||≤𝐶𝛿,(3.18) Hence, we obtain that _ℝ𝑁𝐹𝑣_𝑛𝑑𝑥⟶_ℝ𝑁𝐹(𝑣)𝑑𝑥.(3.19)

Lemma 3.2. The numbers 𝐴 and 𝑐 satisfy the inequality 𝐴≤𝑐.

Proof. For each 𝑣∈𝑊^1,𝑁(ℝ^𝑁)⧵{0}, since we only consider the nontrivial solutions of the problem (1.1), we divide them into two cases to consider.
Case 1. Let 𝑣⁺=max{𝑣,0}≠0, we define the function ℎ∶ℝ→ℝ by ℎ(𝑡)=_ℝ𝑁𝐺(𝑡𝑣)𝑑𝑥=_ℝN𝐹(𝑡𝑣)−𝑡^𝑁𝑣^𝑁𝑁𝑑𝑥.(3.20) By (𝑔₁), we obtain that there exists 𝛿>0,0<𝑐₀<1 such that |𝑠|<𝛿, and ||𝑓(𝑠)||<𝑐₀|𝑠|^𝑁−1.(3.21) Hence ℎ(𝑡)≤_ℝ𝑁^0𝑡𝑣𝑐₀|𝑠|^𝑁−1𝑑𝑥−1𝑁_ℝ^𝑁=𝑐𝑡^𝑁𝑣^𝑁𝑑𝑥₀𝑁∫^ℝ𝑁|𝑡𝑣|^𝑁𝑑𝑥−1𝑁∫^ℝ𝑁|𝑡𝑣|^𝑁𝑑𝑥,(3.22) we obtain that ℎ(𝑡)<0 for 𝑡 small enough. On the other hand, by (𝑔₂), we obtain that ℎ(𝑡)>0 for 𝑡 large enough. In this way, there exists 𝑡₀>0 such that ℎ(𝑡₀)=0, That is, 𝑡₀𝑣∈𝒫. Hence 𝐴≤1𝑁_𝑅^𝑁||∇𝑡₀𝑣||^𝑁𝑑𝑥=𝐼𝑡₀𝑣≤max_𝑡≥0𝐼(𝑡𝑣).(3.23)
Case 2. Let 𝑣⁺=max{𝑣,0}=0, since 𝑓(𝑠)=0 for all 𝑠<0, we obtain max_𝑡≥0𝐼(𝑡𝑣)=+∞.(3.24) As a consequence, 𝐴≤𝑐.(3.25) Combining Cases 1 and 2, we obtain that 𝐴≤𝑐.

Lemma 3.3. The number 𝐴 defined by (2.8) is positive, that is, 𝐴>0.

Proof . Clearly, 𝐴≥0. Assume by contradiction that 𝐴=0 and let {𝑢_𝑛} be a minimizing sequence in 𝑊^1,𝑁(ℝ^ℕ) to 𝐴, that is, 1𝑁_ℝ^𝑁||∇𝑢_𝑛||^𝑁𝑑𝑥⟶𝐴=0with_ℝ𝑁𝐺𝑢_𝑛𝑑𝑥=0.(3.26) For each 𝜆_𝑛>0, set 𝑣_𝑛(𝑥)=𝑢_𝑛(𝑥/𝜆_𝑛) satisfying 1𝑁_ℝ^𝑁||∇𝑣_𝑛||^𝑁𝑑𝑥=1𝑁_ℝ^𝑁||∇𝑣_𝑛||^𝑁𝑑𝑥.(3.27) Similarly, we have _ℝ𝑁𝐺𝑣_𝑛𝑑𝑥=𝜆^𝑁𝑛_ℝ𝑁𝐺𝑢_𝑛𝑑𝑥=0,_ℝ𝑁||𝑣_𝑛||^𝑁𝑑𝑥=𝜆^𝑁𝑛_ℝ𝑁||𝑢_𝑛||^𝑁𝑑𝑥.(3.28) We choose 𝜆^𝑁𝑛=1/∫_ℝ𝑁|𝑢_𝑛|^𝑁𝑑𝑥, so ∫_ℝ𝑁|𝑣_𝑛|^𝑁𝑑𝑥=1. Then we get 1𝑁_ℝ^𝑁||∇𝑣_𝑛||^𝑁𝑑𝑥→𝐴=0,_ℝ𝑁||𝑣_𝑛||^𝑁𝑑𝑥=1,_ℝ𝑁𝐺𝑣_𝑛𝑑𝑥=0.(3.29)
In what follows, we study in the space 𝑊^1,𝑁_rad(ℝ^𝑁). Firstly, we assume that there exists 𝑣∈𝑊^1,𝑁_rad(ℝ^𝑁) such that 𝑣_𝑛⇀𝑣 in 𝑊^1,𝑁_rad(ℝ^𝑁).
On one hand, since (1/𝑁)∫_ℝ𝑁|∇𝑣_𝑛|^𝑁→𝐴=0, then ∃𝑁₀>0, for all 0<𝜖<1, when 𝑛>𝑁₀, we have ∫_ℝ𝑁|∇𝑣_𝑛|^𝑁𝑑𝑥<𝜖<1 and we also know that ∫_ℝ𝑁|𝑣_𝑛|^𝑁𝑑𝑥=1, so ‖𝑣_𝑛‖_𝐿^𝑁_(ℝ^𝑁₎≤𝑀<∞. From Lemma 3.1, we have _ℝ𝑁𝐹𝑣_𝑛𝑑𝑥⟶_ℝ𝑁𝐹(𝑣)𝑑𝑥.(3.30) Note that _ℝ𝑁𝐺𝑣_𝑛𝑑𝑥=_ℝ𝑁𝐹𝑣^𝑛−1𝑁||𝑣^𝑛||^𝑁𝑑𝑥=0,(3.31) so we have _ℝ𝑁𝐹𝑣_𝑛𝑑𝑥=1𝑁_ℝ^𝑁||𝑣_𝑛||^𝑁𝑑𝑥=1𝑁.(3.32) Hence, we have _ℝ𝑁𝐹(𝑣)𝑑𝑥=1𝑁.(3.33)
It implies that 𝑣≠0. On the other hand, since 𝑣_𝑛⇀𝑣 in 𝑊^1,𝑁_rad(ℝ^𝑁), and the space 𝑊^1,𝑁_rad(ℝ^𝑁) is a reflexible Banach space, we have lim_𝑛→∞inf1𝑁_ℝ^𝑁||∇𝑣_𝑛||^𝑁𝑑𝑥≥1𝑁_ℝ^𝑁||∇𝑣||^𝑁𝑑𝑥≥0.(3.34) Since lim_𝑛→∞inf1𝑁_ℝ^𝑁||∇𝑣_𝑛||^𝑁𝑑𝑥=𝐴=0,(3.35) we get 1𝑁_ℝ^𝑁||∇𝑣||^𝑁𝑑𝑥=0.(3.36) From which it follows that 𝑣=0, we have an absurd. Hence, we have 𝐴>0.(3.37)

Lemma 3.4. If 𝜆>(𝑞−𝑁/𝑞)^{(𝑞−𝑁)/𝑁}𝐶^{𝑞𝑞/𝑁}, then 𝑐<1/𝑁.

Proof. From (𝑔₃), we have 𝑓(𝑠)>𝜆𝑠^𝑞−1, for all 𝑠≥0. Now we choose 𝜓∈W^1,𝑁_rad(ℝ^𝑁) such that 𝜓≥0,‖𝜓‖^𝑁𝑞=𝐶^𝑞−1,‖𝜓‖_𝑊^1,𝑁_(ℝ^𝑁₎=1.(3.38) Hence, we have 𝑐≤max_𝑡≥0𝐼(𝑡𝜓)=max_𝑡≥0𝑁1_ℝ^𝑁||∇(𝑡𝜓)||^𝑁=max_𝑡≥0+||𝑡𝜓||^𝑁𝑑𝑥−_ℝ𝑁𝐹(𝑡𝜓)𝑑𝑥𝑡𝑁^𝑁−_ℝ^𝑁^0𝑡𝜓≤max_𝑡≥0𝑓(𝑠)𝑑𝑠𝑑𝑥𝑡𝑁^𝑁−𝜆_ℝ^𝑁^0𝑡𝜓=max_𝑡≥0𝑠^𝑞−1𝑑𝑠𝑑𝑥𝑡𝑁^𝑁−𝜆𝑞𝑡^𝑞_ℝ^𝑁𝜓^𝑞𝑑𝑥.(3.39) Let 𝐾(𝑡)=𝑡^𝑁/𝑁−(𝜆𝑡^𝑞/𝑞)∫_ℝ𝑁𝜓^𝑞𝑑𝑥, then 𝐾(𝑡) is continuous function, we have 𝐾^(𝑡)=𝑡^𝑁−1−𝜆𝑡^𝑞−1_ℝ𝑁𝜓^𝑞𝑑𝑥=0.(3.40) By a simple calculation, when 𝑡₀=(1/𝜆∫_ℝ𝑁𝜓^𝑞𝑑𝑥)^{1/(𝑞−𝑁)}>0, we have max_𝑡>0𝐾(𝑡)=𝐾𝑡₀=1𝑁𝜆∫_ℝ𝑁1𝜓^𝑞𝑑𝑥^{𝑁/(𝑞−𝑁)}−𝜆𝑞_ℝ^𝑁𝜓^𝑞𝑑𝑥1𝜆∫_ℝ𝑁𝜓^𝑞𝑑𝑥^{𝑞/(𝑞−𝑁)}=𝑞−𝑁𝑁𝑞𝜆^{−𝑁/(𝑞−𝑁)}𝐶^{𝑞(𝑞/𝑁)⋅(𝑁/(𝑞−𝑁))}<𝑞−𝑁𝑁𝑞𝑞−𝑞𝑁^{((𝑞−𝑁)/𝑁)⋅(−𝑁/(𝑞−𝑁))}𝐶^{𝑞(𝑞/𝑁)(−𝑁/(𝑞−𝑁))}𝐶^{𝑞𝑞/(𝑞−𝑁)}=1𝑁.(3.41) Hence, we have 𝑐<1𝑁.(3.42)

Lemma 3.5. The number 𝐴 is attained, that is, there exists 𝑢∈𝑊^1,𝑁_rad(ℝ^𝑁) such that 𝐴=∫_ℝ𝑁|∇𝑢|^𝑁𝑑𝑥 and ∫_ℝ𝑁𝐺(𝑢)𝑑𝑥=0.

Proof. Let {𝑢_𝑛} be a minimizing sequence in 𝑊^1,𝑁_rad(ℝ^𝑁) for 𝐴, that is, 1𝑁_ℝ^𝑁||∇𝑢_𝑛||^𝑁𝑑𝑥⟶𝐴(𝑛⟶∞),_ℝ𝑁𝐺𝑢_𝑛𝑑𝑥=0.(3.43) Arguing as in Lemma 3.3, we assume that ∫_ℝ𝑁|𝑢_𝑛|^𝑁𝑑𝑥=1. From (3.43), Lemmas 3.3 and 3.4, we obtain lim_𝑛→∞_ℝ𝑁||∇𝑢_𝑛||^𝑁𝑑𝑥=𝑁𝐴≤𝑁𝑐<1.(3.44) From Lemma 3.1, _ℝ𝑁𝐹𝑢_𝑛𝑑𝑥⟶_ℝ𝑁𝐹(𝑢)𝑑𝑥(3.45) where 𝑢_𝑛⇀𝑢in𝑊^1,𝑁(ℝ^𝑁),as𝑛→∞.
By (3.43) and (3.45), we have _ℝ𝑁𝐹𝑢_𝑛𝑑𝑥=1𝑁_ℝ^𝑁||𝑢_𝑛||^𝑁𝑑𝑥=1_ℝ𝑁𝑁,𝐹(𝑢)𝑑𝑥=1𝑁.(3.46) It implies that 1𝑢≠0,(3.47)𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥≤lim_𝑛→∞inf1𝑁_ℝ^𝑁||∇𝑢_𝑛||^𝑁𝑑𝑥=𝐴,(3.48)_ℝ𝑁|𝑢|^𝑁𝑑𝑥≤lim_𝑛→∞inf_ℝ𝑁||𝑢_𝑛||^𝑁𝑑𝑥=1.(3.49) From (3.48) and (3.49), we have _ℝ𝑁𝐺(𝑢)𝑑𝑥=_ℝ𝑁𝐹(𝑢)𝑑𝑥−1𝑁_ℝ^𝑁|𝑢|^𝑁𝑑𝑥=1𝑁−𝑁1_ℝ^𝑁|𝑢|^𝑁𝑑𝑥≥0.(3.50) If ∫_ℝ𝑁𝐺(𝑢)𝑑𝑥≠0, from (3.50), we have ∫_ℝ𝑁𝐺(𝑢)𝑑𝑥>0. Consider the function ℎ defined in Lemma 3.2 relative to the function: ℎ(𝑡)=_ℝ𝑁𝐺(𝑡𝑢)𝑑𝑥.(3.51) We concludes that ℎ(𝑡)<0 for 𝑡 small enough. On the other hand, ℎ(1)=∫_ℝ𝑁𝐺(𝑢)𝑑𝑥>0. In this way, we obtain that there is 𝑡₀∈(0,1) such that ℎ(𝑡₀)=0, that is, _ℝ𝑁𝐺𝑡₀𝑢𝑑𝑥=0.(3.52) Hence, from (3.48), 0<1𝑁_ℝ^𝑁||∇𝑡₀𝑢||^𝑁𝑑𝑥=1𝑁𝑡^𝑁0_ℝ𝑁||∇𝑢||^𝑁𝑑𝑥≤𝑡^𝑁0𝐴<𝐴.(3.53) However, from (3.52), we have 𝑡₀𝑢∈𝒫. Hence, we obtain 1𝑁_ℝ^𝑁||∇𝑡₀𝑢||^𝑁𝑑𝑥≥𝐴.(3.54) Which is contradictory with (3.53).
Thus, we obtain _ℝ𝑁𝐺(𝑢)𝑑𝑥=0.(3.55) It implies 𝑢∈𝒫 and 1𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥≥𝐴.(3.56) From (3.48) and (3.56), we obtain that 1𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥=𝐴,(3.57) with ∫_ℝ𝑁𝐺(𝑢)𝑑𝑥=0,𝑢≠0.
We obtain that 𝐴 is attained.

Proof of Theorem 1.2. From Lemma 3.5, there is 𝑢∈𝑊^1,𝑁_rad(ℝ^𝑁)⧵{0} such that 1𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥=𝐴,_ℝ𝑁𝐺(𝑢)𝑑𝑥=0.(3.58) we will prove that 𝑚=𝑏=𝐴.
By Lagrange multipliers, there exists 𝜌∈ℝ, such that _ℝ𝑁||∇𝑢||^𝑁−2∇𝑢∇𝑣𝑑𝑥=𝜌_ℝ𝑁𝑔(𝑢)𝑣𝑑𝑥,(3.59) for every 𝑣∈𝑊^1,𝑁(ℝ^𝑁).
Define the rescaled function 𝑢_𝜌1/𝑁=𝑢(𝜌^−1/𝑁𝑥), which is a nontrivial solution of (1.1) with _ℝ𝑁||∇𝑢_𝜌1/𝑁||^𝑁𝑑𝑥=_ℝ𝑁||∇𝑢||^𝑁𝑑𝑥,_ℝ𝑁𝐺𝑢_𝜌1/𝑁𝑑𝑥=𝜌_ℝ𝑁𝐺(𝑢)𝑑𝑥=0.(3.60) Thus, we have 𝑚≤𝐼𝑢_𝜌1/𝑁=1𝑁_ℝ^𝑁||∇𝑢_𝜌1/𝑁||^𝑁𝑑𝑥−_ℝ𝑁𝐺𝑢_𝜌1/𝑁𝑑𝑥=1𝑁_ℝ^𝑁||∇𝑢||^𝑁𝑑𝑥=𝐴.(3.61) So, we have 𝑚≤𝐴.(3.62)
For each 𝛾∈Γ, one has 𝛾([0,1])∩𝒫≠∅ from [4]. We obtain that there exists 𝑡₀∈[0,1] such that 𝛾(𝑡₀)∈𝒫, that is, 𝛾(𝑡₀) satisfied that ∫_ℝ𝑁𝐺(𝛾(𝑡₀))𝑑𝑥=0 and then 𝐴≤1𝑁_ℝ^𝑁||∇𝛾𝑡₀||^𝑁𝑑𝑥−1𝑁_ℝ^𝑁𝐺𝛾𝑡₀𝑑𝑥=𝐼𝛾𝑡₀.(3.63) Hence 𝐴≤𝐼(𝛾(𝑡₀))≤max_𝑡∈[0,1]𝐼(𝛾(𝑡)) for every 𝛾∈Γ, we obtain that 𝐴≤𝑏.(3.64) From (3.62) and (3.64), we obtain that 𝑚≤𝐴≤𝑏.
On the other hand, for every nontrivial solution 𝜔∈𝑊^1,𝑁(ℝ^𝑁) of the problem (1.1), there exists a path 𝛾_𝜔∈Γ such that 𝜔∈𝛾_𝜔([0,1]) and max_𝑡∈[0,1]𝐼(𝛾_𝜔(𝑡))=𝐼(𝜔). Consequently, 𝑏≤𝐼(𝜔),𝑏≤𝑚.
In conclusion, we obtain 𝑚=𝐴=𝑏.(3.65) Hence, the function 𝑢_𝜌1/𝑁 is a ground-state solution of the problem (1.1).

Acknowledgment

This paper was supported by Shanghai Natural Science Foundation under Project: 11ZR1424500.