Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity

Qihan He; Juntao Lv; Zongyan Lv

doi:10.1515/anona-2020-0168

Open Access Published by De Gruyter March 5, 2021

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|^p–2t at infinity

Qihan He , Juntao Lv and Zongyan Lv

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0168

Abstract

We consider the following p-harmonic problem

Δ(|Δu|p−2Δu)+m|u|p−2u=f(x,u),x∈RN,u∈W2,p(RN),

where m > 0 is a constant, N > 2p ≥ 4 and limt→∞f(x,t)|t|p−2t=l uniformly in x, which implies that f(x, t) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f(x, u) ≡ f(u), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

Keywords: p-harmonic equation; nontrivial solutions; Ambrosetti-Rabinowitz condition

MSC 2010: 35J35; 35J60; 47J30

1 Introduction

In this paper, we deal with the existence and nonexistence of nontrivial solutions to the following p-harmonic problem

Δ(|Δu|p−2Δu)+m|u|p−2u=f(x,u), x∈RN,u∈W2,p(RN), (1.1)

where m > 0 denotes a constant and N > 2p ≥ 4.

Throughout this paper, we assume that f(x, t) satisfies the following conditions:

f : ℝ^N × ℝ → ℝ satisfies the Caratheodory conditions; f(x, t)t ≥ 0, ∀ (x, t) ∈ ℝ^N × ℝ and f(x, 0) ≡ 0, ∀ x ∈ ℝ^N.
limt→0f(x,t)|t|p−2t=0 uniformly in x ∈ ℝ^N.
limt→∞f(x,t)|t|p−2t=l uniformly in x ∈ ℝ^N for some l ∈ (0, +∞).
For a.e. x ∈ ℝ^N, f(x,t)|t|p−2t is nondecreasing with respect to t > 0, and nonincreasing with respect to t < 0.
∃ f̄(t) ∈ C¹(ℝ) such that |f(x, t)| ≥ |f̄(t)|, ∀ (x, t) ∈ ℝ^N × ℝ, meas {x ∈ ℝ^N : |f(x, t)| > |f̄(t)|} > 0, ∀t ∈ ℝ ∖ {0} and lim|x|→+∞ f(x, t) = f̄(t) uniformly in t ∈ ℝ.
f̄(t) ∈ C¹(ℝ) satisfies that (p – 1)f̄(t) > f̄′(t)t for all t < 0 and (p – 1)f̄(t) < f̄′(t)t for all t > 0.

Definition 1.1

We call u ∈ W^2,p(ℝ^N) a (weak) solution of (1.1) if for all ϕ ∈ W^2,p(ℝ^N), we have

∫RN(|Δu|p−2ΔuΔϕ+m|u|p−2uϕ)dx=∫RNf(x,u)ϕdx.

It is easy to see that any solution of (1.1) corresponds to a critical point of the following energy functional defined on W^2,p(ℝ^N):

I(u)=1p∫RN(|Δu|p+m|u|p)dx−∫RNF(x,u)dx, (1.2)

where F(x, u) = ∫0u f(x, s)ds.

The famous Mountain Pass Theorem proposed in [1] and the constraint minimization are the useful tools to get critical points of I(u). Based on the Mountain Pass Theorem, many results about the existence of solutions to second order nonlinear elliptic problems included p-laplacian or bi-harmonic operators have been obtained (see [2, 3, 4, 5] and the references therein).

Evidently, compared with the case of bounded domain, when treating a problem in ℝ^N, the compactness of Sobolev imbedding is absent. Researchers have attempted various methods to study this kind of problem. For example, to the following semilinear elliptic problem

−Δu+mu=f(x,u), x∈RN,u∈H1(RN), (1.3)

where f(x, u) is spherical symmetrical or autonomous , the existence of nontrivial solutions to (1.3) was considered in the radially symmetrical Sobolev space (see [6, 7, 8]). However, this method can not be used to the case of general f(x, t). To dealt with this kind of problem, the concentration-compactness principle was proved by P. L. Lions. Many researchers have studied the variational elliptic problems in ℝ^N by this principle (see [9, 10] and the references therein).

Yang and Zhu [11] considered the following quasilinear elliptic equation

−div(|∇u|p−2∇u)+a(x)|u|p−2u=f(x,u), x∈RN,u∈W1,p(RN), (1.4)

where N > p ≥ 2 and a(x) satisfies the following assumption:

(a) a(x) ∈ C(ℝ^N), a(x) ≥ a₀ > 0 and lim|x|→+∞ a(x) = ā > 0.

Deng, Gao and Jin [12] studied the p-harmonic problem (1.1). The authors in [12] required that f(x, t) is subcritical and satisfies some assumptions similar to (C₁) – (C₂), (C₄) – (C₆) and the (AR) condition:

∃θ>0 s.t 0≤F(x,t)=∫0tf(x,s)ds≤1p+θf(x,t)t, ∀ (x,t)∈RN×R. (1.5)

This condition implies that for some C > 0,

F(x,t)≥C|t|p+θfor |t|>0 large.

However, (C₃) implies that f(x, t) is asymptotic to |t|^p–2t at infinity, and (1.5) does not satisfy. During the last twenty years, Researchers have shown a lot of results about (1.3) and (1.4) without the (AR) condition(see [13, 14, 15, 16, 17]). Using the concentration-compactness principle together with a variant version of Mountain Pass Theorem, Li and Zhou in [18] proved that (1.3) has a positive solution under similar conditions on f(x, t) as (C₁) – (C₆). After that, He and Li in [19] proved the existence of a nontrivial solution to the p & q-Laplacian equation under similar assumptions on f(x, t) as (C₁) – (C₆). With the help of Ekeland variational principle and Mountain Pass Theorem, the existence of multiple solutions for boundary value problem of nonhomogeneous p-harmonic equation has been proved (see [20, 21] and the references therein).

This paper is motivated by [12], [18] and [19] . We want to consider the existence and nonexistence of nontrivial solutions to the equation problem (1.1). To our best knowledge, there are few results on problem (1.1) when f(x, t) is asymptotic to |t|^p–2t at infinity.

We first define the limited problem of (1.1) as follows:

Δ(|Δu|p−2Δu)+m|u|p−2u=f¯(u), x∈RN,u∈W2,p(RN). (1.6)

For any u ∈ W^2,p(ℝ^N), we define

I∞(u)=1p∫RN(|Δu|p+m|u|p)dx−∫RNF¯(u)dx, (1.7)

where F¯(u)≜∫0uf¯(s)ds. Clearly, I^∞ ∈ C¹(W^2,p(ℝ^N), ℝ). Denote

Λ={u∈W2,p(RN):〈I∞′(u),u〉=0,u ̸≡0}, (1.8)

where 〈 ⋅, ⋅〉 denotes the dual paring between W^2,p(ℝ^N) and (W^2,p(ℝ^N))^–1. Λ ≠ ∅ will be shown in Lemma 2.4 below. So

J∞=inf{I∞(u):u∈Λ} (1.9)

is well-defined.

Our main results can be stated as follows:

Theorem 1.2

Assume that conditions (C₁) – (C₆) hold. If l ∈ (m, +∞), then J^∞ > 0 and it is achieved by some ũ ∈ W^2,p(ℝ^N) ∖ {0}, which is a ground state solution for problem (1.6). Moreover, if l ≤ m, then problem (1.6) has no nontrivial solutions.

Theorem 1.3

Assume that (C₁) – (C₆) hold. Then there exists at least a nontrivial weak solution to (1.1) if l ∈ (m, +∞) and there is no nontrivial weak solutions to (1.1) if l ≤ m.

To show the Theorems, we need to prove the following Proposition:

Proposition 1.4

(Pohozaev identity for weak solution) Suppose that f̄ is a continuous function satisfying the following growth condition:

|f¯(t)|≤c|t|p−1+C|t|d0−1 for all t∈R,

where c, C > 0 and d0=Np−pN−2p. If u is a weak solution of the p-harmonic problem (1.6), that is

∫RN(|Δu|p−2ΔuΔϕ+m|u|p−2uϕ)dx=∫RNf¯(u)ϕdx, ∀ϕ∈W2,p(RN), (1.10)

then u satisfies the Pohozaev type identity:

N−2pNp∫RN|Δu|pdx+1p∫RNm|u|pdx=∫RNF¯(u)dx. (1.11)

Remark 1.5

The role of exponent d₀ = Np−pN−2p is to guarantee that (2.7) is true. It is easy to verify that p^* := NpN−p < d₀ < NpN−2p =: p^** and (2.7) holds naturally for d₀ = p^** if the weak solution u ∈ LLoc∞ (ℝ^N). So the Pohozaev type identity (1.11) is also true for d₀ = p^** if u ∈ LLoc∞ (ℝ^N).

Following from the Proposition 1.4, we can get the following Corollary directly.

Corollary 1.6

Assume that u ∈ W^2,p(ℝ^N) is a weak solution of

Δ(|Δu|p−2Δu)=λ|u|q−2u, x∈RN

where λ ∈ ℝ and q ∈ [p, d₀]. Then u ≡ 0.

In the proofs of our main results, we are faced with several difficulties:

Firstly, the method in [19] can not be applied directly to the p-harmonic problem. For example, for the quasilinear elliptic problem (1.4), u ∈ W^1,p(ℝ^N) implies that |u|, u⁺, u^– ∈ W^1,p(ℝ^N), where u⁺ = max{u, 0}, u^– = max{–u, 0}. From this fact, we can prove that a solution can be taken to be positive. While for the p-harmonic problem (1.1), this way fails completely since u ∈ W^2,p(ℝ^N) does not imply that |u|, u⁺, u^– ∈ W^2,p(ℝ^N). To this end, we need to take prolongation on f(x, t) for t < 0.

Secondly, as we can see in [18], the Pohozaev identity played an important role in the proofs of the main results, which is slightly different from what [19] did. However, we can not get the Pohozaev identity of p-harmonic problem (1.6) as usual, since there is less information on the regularity of solutions to (1.6). Due to the lack of regularity of the solutions to problem (1.6), the usual method to derive the corresponding Pohozaev identity can not work. Inspired by [22], we take ϕ:=ψ∑j=1NxjDjhu as a test function to derive the corresponding Pohozaev identity, which relaxs the restriction on the regularity of the solution and where Djhu denotes the difference quotient and ψ is a given cut-off function. But our problem (1.6) is a quasilinear elliptic equation of fourth order or more, we need to estimate the higher order difference. Thus more delicate analysis is needed.

Thirdly, W^2,p(ℝ^N) is not a Hilbert space in general. It is not clear that, up to a subsequence,

|Δun|p−2Δun⇀|Δu|p−2Δu in Lpp−1(RN),

even if the (PS) sequence u_n} of the functional I is bounded in W^2,p(ℝ^N) and

un⇀uin W2,p(RN).

Hence, more delicate analysis is needed to prove that

Δun→Δu a.e. in RN.

Finally, since f(x, t) and f̄(t) are asymptotic to |t|^p–2t at infinity, the (AR) condition (1.5) does not satisfy so that showing the boundedness of any (PS)_c sequence for I(u) or I^∞(u) in W^2,p(ℝ^N) has become one of the main difficulties for studying the existence of nontrivial weak solutions to (1.1) or (1.6) in W^2,p(ℝ^N). To show Theorem 1.2, inspired by [18], we apply Ekeland’s variational principle to get a minimizing sequence {u_n} for J^∞ with I^∞^′(u_n) → 0 in (W^2,p(ℝ^N))^–1, which guarantees that we can show J^∞ > 0. Then we can show that J^∞ is achieved by some u₀. As to Theorem 1.3, motivated by [19], we would prove it by a mountain pass theorem without Cerami condition together with the concentration-compacteness principle. With the help of the ground state solution ũ to (1.6) obtained in Theorem 1.2, we construct the mountain pass level c as

c=infγ∈Γmax0≤t≤1I(γ(t)),

where Γ = {γ ∈ C([0, 1], W^2,p(ℝ^N)) : γ(0) = 0, γ(1) = t₀ũ} for some t₀ > 0 large enough and which is slightly different from what [18] did. Liu and Zhou [18] use u~(xt0) instead of t₀ũ in the definition of Γ and prove that I( u~(xt0) ) < 0 for t₀ large enough. But for the solutions of (1.6), we can not obtain the regularity result to ensure that γ0(t)=u~(xtt0), t∈(0,1],0, t=0, belongs to Γ, which is a key point to show c < J^∞. So the method in [18] does not work here. Due to limt→∞f¯(t)|t|p−2t=l, (C₃) and (C₅), we can show that I(tũ) < I^∞(tũ) < 0 for t > 0 large enough, which implies that the mountain pass level c defined above is well-defined and c < J^∞. By the mountain pass theorem without the Cerami condition, we can see that there exists a Cerami sequence {u_n} ⊂ W^2,p(ℝ^N) of I(u) at the level c, that is

I(un)→cand(1+∥un∥)∥I′(un)∥(W2,p(RN))−1→0,as n→+∞. (1.12)

Then we can apply the fact that c < J^∞ and the concentration-compactness principle to prove that {u_n} is bounded in W^2,p(ℝ^N) and the weak limit of such subsequence of {u_n} is a nontrivial weak solution of (1.1).

This paper is organized as follows. In Section 2, we first derive the Pohozaev identity for the weak solutions of problem (1.6), and some preliminary lemmas are presented. The proof of Theorem 1.2 is put into the Section 3. In Section 4, by using a variant version of Mountain Pass Theorem, we devote to prove Theorem 1.3.

2 Some notations and preliminary Lemmas

In this section, we devote to some notations and preliminary Lemmas, which are crucial in our proofs of main results.

In the sequel, C represents positive constant. We denote the norm of u ∈ {L^p(ℝ^N) by

∥u∥p=(∫RN|u|pdx)1p,1≤p<+∞,

and the norm of u ∈ W^2,p(ℝ^N) by

∥u∥=[∫RN(|Δu|p+m|u|p)dx]1p,2≤p<+∞.

Let N > 2p and denote p^** = NpN−2p . It follows from (C₁) – (C₅) that for any ε > 0, τ ∈ (p, p^**), there exists C_ε > 0 such that for all x ∈ ℝ^N, t ∈ ℝ,

|f(x,t)|≤ε|t|p−1+Cε|t|τ−1, |f¯(t)|≤ε|t|p−1+Cε|t|τ−1, (2.1)

F(x,t)≤ε|t|p+Cε|t|τ, F¯(t)≤ε|t|p+Cε|t|τ, (2.2)

|f(x,t)|≤l|t|p−1, |f¯(t)|≤l|t|p−1, (2.3)

where F¯(t)=∫0tf¯(s)ds. Combining (C₂), (C₃) and (C₅), we see that

limt→0f¯(t)|t|p−2t=0, limt→∞f¯(t)|t|p−2t=l. (2.4)

According to (C₆), ones can see that

f¯(t)|t|p−2t is strictly increasing in t>0, and strictly decreasing in t<0,F¯(t)<1pf¯(t)t, ∀t≠0. (2.5)

Based on the above observations, we are going to prove Proposition 1.4.

The proof of Proposition 1.4

Set g(u) = f̄(u) – m|u|^p–2u, and G(u) = ∫0u g(t)dt. Denote Dihu(x)=u(x+hei)−u(x)h, where h ∈ ℝ∖{0} and e_i denotes the unit vector along coordinate x_i. We take a cut-off function ψ ∈ C0∞ (ℝ^N) satisfying that ψ(x) = 1 for |x| ≤ R, ψ(x) = 0 for |x| ≥ 2R, |∇ψ| ≤ 2R and |Δψ| ≤ 2R2 . Following the idea in [22], we set ϕ=ψ∑j=1NxjDjhu, Taking such ϕ as a test function in ((1.10), we have

∫RN|Δu|p−2ΔuΔ(ψ∑j=1NxjDjhu)dx=∫RNg(u)ψ∑j=1NxjDjhudx. (2.6)

A straightforward computation gives us

∫RN|Δu|p−2ΔuΔ(ψ∑j=1NxjDjhu)dx=∑j=1N∫RN|Δu|p−2ΔuΔ(ψxjDjhu)dx=∑j=1N∫RN|Δu|p−2ΔuDjhuΔ(ψxj)dx+∑j=1N∫RN|Δu|p−2ΔuΔ(Djhu)ψxjdx+2∑i,j=1N∫RN|Δu|p−2ΔuDi(ψxj)Di(Djhu)dx≜I1+I2+I3,

where D_iu denotes ∇u ⋅ e_i. Now we estimate the three terms I₁, I₂ and I₃.

I1 =∑j=1N∫RN|Δu|p−2ΔuDjhuΔ(ψxj)dx=∑j=1N∫RN|Δu|p−2ΔuDjhu(Δψxj+2Djψ)dx.

Since ∥Djhu∥p≤C∥Du∥p, we have Djh u ⇀ D_ju weakly in L^p(ℝ^N), which means that, as h → 0,

I1→∫RN|Δu|p−2Δu(∇u,x)Δψdx+2∫RN|Δu|p−2Δu(∇u,∇ψ)dx.

I2 =∑j=1N∫RN|Δu|p−2ΔuΔ(Djhu)ψxjdx=1p∑j=1N∫RNDjh(|Δu|p)ψxjdx−1p∑j=1N∫RN(Djh(|Δu|p)−p|Δu|p−2ΔuΔ(Djhu))ψxjdx=I21−I22

and

It follows from ∥Djh(ψxj)∥p≤C∥Dj(ψxj)∥p that Djh (ψx_j) ⇀ D_j(ψx_j) weakly in L^p(ℝ^N), which implies that when h → 0,

I21 →−1p∑j=1N∫RN|Δu|pDj(ψxj)dx=−Np∫RN|Δu|pψdx−1p∫RN|Δu|p(∇ψ,x)dx.

Since u satisfies ((1.10), we have

I22 =1p∑j=1N∫RN(Djh(|Δu|p)−p|Δu|p−2ΔuΔ(Djhu))ψxjdx=1p∑j=1N∫RN|Δu(x+hej)|p−|Δu(x)|phψxjdx−1p∑j=1N∫RNp|Δu|p−2ΔuΔu(x+hej)−Δu(x)hψxjdx=1p∑j=1N∫RNp|Δu~|p−2Δu~(Δu(x+hej)−Δu(x))hψxjdx−1p∑j=1N∫RNp|Δu|p−2ΔuΔu(x+hej)−Δu(x)hψxjdx

=∑j=1N∫RN|Δu~|p−2Δu~[Δ(u(x+hej)ψxjh)−u(x+hej)Δ(ψxj)h]dx−∑j=1N∫RN|Δu~|p−2Δu~[Δ(u(x)ψxjh)−u(x)Δ(ψxj)h]dx−2∑i,j=1N∫RN|Δu~|p−2Δu~Di(Djhu)Di(ψxj)dx−∑j=1N∫RN|Δu|p−2Δu[Δ(u(x+hej)ψxjh)−u(x+hej)Δ(ψxj)h]dx+∑j=1N∫RN|Δu|p−2Δu[Δ(u(x)ψxjh)−u(x)Δ(ψxj)h]dx+2∑i,j=1N∫RN|Δu|p−2ΔuDi(Djhu)Di(ψxj)dx=∑j=1N∫RN(g(u~)−g(u))Djhuψxjdx+∑j=1N∫RN(|Δu|p−2Δu−|Δu~|p−2Δu~)DjhuΔ(ψxj)dx+2∑i,j=1N∫RN(|Δu|p−2Δu−|Δu~|p−2Δu~)Di(Djhu)Di(ψxj)dx,

where u͠ = λu + (1 – λ)u(⋅ + he_j), λ ∈ [0, 1]. By the Hölder inequality, we have

|I22| ≤∑j=1N(∫RN|Djhu|pdx)1p(∫RN|g(u~)−g(u)|q|ψxj|qdx)1q+∑j=1N(∫RN|Djhu|pdx)1p(∫RN|Δ(ψxj)|q||Δu|p−2Δu−|Δu~|p−2Δu~|qdx)1q+2∑i,j=1N(∫RN|Di(Djhu)|pdx)1p(∫RN|Di(ψxj)|q||Δu|p−2Δu−|Δu~|p−2Δu~|qdx)1q

Note that ∥Djhu∥p≤C∥∇u∥p<+∞,∥Di(Djhu)∥p≤C∥Δu∥p<+∞, then we have

|I22|≤C(∫RN|g(u~)−g(u)|q|ψxj|qdx)1q+C(∫RN|Δ(ψxj)|q||Δu|p−2Δu−|Δu~|p−2Δu~|qdx)1q+C(∫RN|Di(ψxj)|q||Δu|p−2Δu−|Δu~|p−2Δu~|qdx)1q→0, as h→0,

where 1p+1q=1. Additionally,

I3 =2∑i,j=1N∫RN|Δu|p−2ΔuDi(ψxj)Di(Djhu)dx=2∑i,j=1N∫RN|Δu|p−2ΔuDi(Djhu)(Diψxj+ψδij)dx,

where δ_ij is the Kronecker symbol, that is δ_ij = 1 when i = j and 0 otherwise. By ∥Di(Djhu)∥p≤C∥Di(Dju)∥p, we obtain D_i( Djh u) ⇀ D_i(D_ju) weakly in L^p(ℝ^N), which means that

I3→2∑j=1N∫RN|Δu|p−2Δu(∇ψ,∇(Dju))xjdx+2∫RN|Δu|pψdx.

Next we turn to estimate the right hand side of (2.6).

Since u ∈ W^2,p(ℝ^N), D_iu ∈ L^d(ℝ^N) and u ∈ L^q(ℝ^N), where d ∈ [p, NpN−p ], q ∈ [p, NpN−2p ]. It follows from the definition of Dihu(x), we have ||Dihu(x)||Ld(RN)≤C||Diu||Ld(RN). Therefore, Dihu(x)⇀Diu weakly in L^d(ℝ^N), which implies that, as h → 0,

∫RN|g(u)ψxi(Dihu(x)−Diu)| dx≤∫RN2R(c|u|p−1+C|u|d0−1)|Dihu(x)−Diu| dx≤C∫RN|u|p−1|Dihu(x)−Diu| dx+C∫RN|u|d0−1|Dihu(x)−Diu| dx→0, (2.7)

where we have used the facts that |u|p−1∈Lpp−1(RN)=(Lp(RN))−1,d0=Np−pN−2p and |u|d0−1∈LNpNp−N+p(RN)=(LNpN−p(RN))−1. So we have

∫RNg(u)ψ∑j=1NxjDjhudx →∫RNg(u)ψ(∇u,x)dx=∫RNψ(∇G(u),x)dx=∫RNdiv(G(u)ψx)dx−∫RNG(u)(∇ψ,x)dx−N∫RNG(u)ψdx=−∫RNG(u)(∇ψ,x)dx−N∫RNG(u)ψdx.

Altogether, as h → 0, by (2.6), we have

∫RN|Δu|p−2Δu(∇u,x)Δψdx+2∫RN|Δu|p−2Δu(∇u,∇ψ)dx−Np∫RN|Δu|pψdx−1p∫RN|Δu|p(∇ψ,x)dx+2∑j=1N∫RN|Δu|p−2Δu(∇ψ,∇(Dju))xjdx+2∫RN|Δu|pψdx=−∫RNG(u)(∇ψ,x)dx−N∫RNG(u)ψdx.

Thus

N−2pp∫RN|Δu|pψdx+1p∫RN|Δu|p(∇ψ,x)dx=∫RNG(u)(∇ψ,x)dx+N∫RNG(u)ψdx+∫RN|Δu|p−2Δu(∇u,x)Δψdx+2∫RN|Δu|p−2Δu(∇u,∇ψ)dx+2∑j=1N∫RN|Δu|p−2Δu(∇ψ,∇(Dju))xjdx.

Notice that |∇ψ| ≤ 2R , |Δψ| ≤ 2R2 and supp ∇ψ, supp △ψ ⊂ ⊂ {x ∈ ℝ^N : R ≤ |x| ≤ 2R}, then

∫RN|Δu|p(∇ψ,x)dx≤4∫R≤|x|≤2R|Δu|pdx→0 as R→+∞.

Similarly we have

∫RNG(u)(∇ψ,x)dx→0 as R→+∞,∫RN|Δu|p−2Δu(∇u,x)Δψdx→0 as R→+∞,∫RN|Δu|p−2Δu(∇u,∇ψ)dx→0 as R→+∞,

and

∑j=1N∫RN|Δu|p−2Δu(∇ψ,∇(Dju))xjdx≤4(∫R≤|x|≤2R|Δu|pdx)p−1p(∫R≤|x|≤2R|∇(∇u)|pdx)1p→0 as R→+∞.

On the other hand, as R → +∞,

∫RN|Δu|pψdx→∫RN|Δu|pdx, ∫RNG(u)ψdx→∫RNG(u)dx.

Therefore we obtain the Pohozaev identity

N−2pp∫RN|Δu|pdx=N∫RNG(u)dx,

which gives (1.11) since G(u) = F̄(u) – 1pm|u|p. □

We list some useful Lemmas.

Lemma 2.1

(Lemma 1.1 of [9]) Let {ρ_n} ⊂ L¹(ℝ^N) be a bounded sequence and ρ_n ≥ 0, then there exists a subsequence, still denoted by {ρ_n}, such that one of the following two possibilities occurs:

(Vanishing): limn→+∞supy∈RN∫BR(y)ρn(x)dx=0 for all 0 < R < +∞.
(Nonvanishing): There exist α > 0, 0 < R < +∞ and {y_n} ⊂ ℝ^N such that

limn→+∞∫BR(yn)ρn(x)dx≥α>0.

Lemma 2.2

(Lemma 2.1 of [12]) Let p ≥ 2, p ≤ τ < p^** = NpN−2p . Assuming that {u_n} is bounded in W^2,p(ℝ^N) and as n → +∞

supy∈RN∫BR(y)|un|τdx→0 for some R>0,

then u_n → 0 in L^α(ℝ^N) as n → +∞, for any α ∈ (p, p^**).

Lemma 2.3

For the functional I^∞ defined by (1.7), if u_n ⊂ W^2,p(ℝ^N) satisfies 〈I^{∞^′}(u_n), u_n〉 = 0 for all n ≥ 1, then I^∞(tu_n) ≤ I^∞(u_n) for all t > 0.

Proof

A similar proof can be found in [19]. We omit it here.□

In the following, we give several results for the set Λ and the minimization problem (1.9).

Lemma 2.4

Assume that (C₁) – (C₃) hold and l > m. Then Λ ≠ ∅.

Proof

Let

[d(N)]p=∫RNe−p|x|pdx.

For α > 0, we set

ωα(x)=[d(N)]−1αNp2e−α|x|p

and

D(N)=pp[d(N)]−p∫RN[p|x|2p−2e−|x|p−(p−2+N)|x|p−2e−|x|p]pdx.

It is easy to see that ω_α ∈ W^2,p(ℝ^N). Straightforward calculations show that

∥ωα∥p=1, ∥Δωα∥pp=α2D(N).

By using (2.1), for some τ ∈ (p, p^**), we have

〈I∞′(tωα),tωα〉 =tp∥ωα∥p−∫RNf¯(tωα)tωαdx≥tp∥ωα∥p−∫RNεtp|ωα|pdx−∫RNCεtτ|ωα|τdx=tp(α2D(N)+m−ε)−Cεtτ∫RN|ωα|τdx>0, (2.8)

if we take t > 0 small enough. On the other hand, if we choose α∈(0,[l−mD(N)]12), where l is given by (C₃), then

∥Δωα∥pp<l−m. (2.9)

whence, by Fatou’s lemma together with (2.4), one has

limt→+∞〈I∞′(tωα),tωα〉tp =∥ωα∥p−limt→+∞∫RNf¯(tωα)tωαtpdx≤∥ωα∥p−∫RNlimt→+∞f¯(tωα)ωαp(tωα)p−1dx=∥Δωα∥pp+m−l<0.

So 〈I^{∞^′}(tω_α), tω_α〉 → –∞, as t → +∞, which, together with (2.8), implies that there exists a t^* > 0 such that 〈I^{∞^′}(t^*ω_α), t^*ω_α〉 = 0. that is t^*ω_α ∈ Λ.□

Now we continue to study Λ. By f̄(t) ∈ C¹(ℝ), it is easy to see that the functional g(u) ≜ 〈I^{∞^′}(u), u〉 = ∫_ℝ^N(|Δu|^p + m|u|^p)dx – ∫_ℝ^N f̄(u)udx ∈ C¹( W^2,p(ℝ^N), ℝ). Then by (C₆), for any u ∈ Λ

〈g′(u),u〉=p∫RN(|Δu|p+m|u|p)dx−∫RN[f¯(u)u+f¯′(u)u2]dx=∫RN[(p−1)f¯(u)u−f¯′(u)u2]dx=∫RN[(p−1)f¯(u)−f¯′(u)u]udx<0.

So Λ is a closed complete submanifold of W^2,p(ℝ^N) with the natural Finsler structure (see [23]). Therefore, using the standard Ekeland’s variational principle on Finsler manifold (see Lemma 2.4 in [18] and Corollary 1.3 in Chapter 2 of [23]) and Lemma 2.4, we can deduce the following Lemma.

Lemma 2.5

If (C₁) – (C₃) and (C₅) – (C₆) hold, then there exists a sequence {u_n} ⊂ Λ such that as n → +∞

I∞(un)→J∞=infu∈ΛI∞(u) and I∞′(un)→0 in (W2,p(RN))−1.

Lemma 2.6

Assume that (C₁) – (C₃) and (C₅) – (C₆) hold. Then we have that J^∞ > 0.

Proof

If J^∞ = 0, then it follows from Lemma 2.5 that there exists a sequence {u_n} ⊂ Λ such that as n → +∞

I∞(un)=1p∥un∥p−∫RNF¯(un)dx→J∞=0, (2.10)

〈I∞′(un),un〉=∥un∥p−∫RNf¯(un)undx=0, (2.11)

and

I∞′(un)→0 in (W2,p(RN))−1. (2.12)

First, we show the boundedness of {u_n} in W^2,p(ℝ^N). Seeking a contradiction, we assume that

∥un∥→+∞, as n→+∞, (2.13)

and for any fixed α > 0, let

tn=α∥un∥, ωn(x)=tnun(x)=αun(x)∥un(x)∥. (2.14)

Clearly {ω_n} is bounded in W^2,p(ℝ^N). For ρ_n = |ω_n|^p, then {ρ_n} ⊂ L¹(ℝ^N) is a bounded sequence and ρ_n ≥ 0, which implies that either (i) or (ii) of Lemma 2.1 holds. We will get contradictions by showing none of these alternatives can occur.

Vanishing: In this case, it follows from Lemma 2.2 and (2.2) that

∫RNF¯(ωn)dx→0, as n→+∞.

Then by (2.14), we have

I∞(ωn)=1p∥ωn∥p−∫RNF¯(ωn)dx=1p∥ωn∥p+on(1)=1pαp+on(1), (2.15)

where o_n(1) → 0 as n → +∞. But by Lemma 2.3 and (2.10), one has

I∞(ωn)=I∞(tnun)≤I∞(un)→J∞=0, as n→+∞, (2.16)

which contradicts to (2.15).
Nonvanishing: In this case, there exist η > 0, R > 0 and {y_n} ⊂ ℝ^N such that

limn→+∞∫BR(yn)|ωn|pdx≥η>0. (2.17)

Set ω̃_n(x) = ω_n(x + y_n), then ∥ω̃_n∥ = ∥ω_n∥ = α and by Sobolev imbedding, we may assume that for some ω̃ ∈ W^2,p(ℝ^N), such that as n → +∞

ω~n⇀ω~ in W2,p(RN),ω~n→ω~ in Llocp(RN),ω~n→ω~ a.e. in RN.

It follows from (2.17) that

ω~≢0. (2.18)

By (2.5), for n large enough, we have

tn=α∥un∥∈(0,1) and f¯(tnun)|tnun|p−2tnun≤f¯(un)|un|p−2un.

Hence from (2.11)

∥ωn∥p−∫RNf¯(ωn)ωndx =tnp∥un∥p−∫RNf¯(tnun)tnundx=tnp[∥un∥p−∫RNf¯(tnun)|tnun|p−2tnun|un|pdx]≥tnp[∥un∥p−∫RNf¯(un)|un|p−2un|un|pdx]=0.

Then Fatou’s Lemma and (2.5) imply that

I∞(ωn) =1p∥ωn∥p−∫RNF¯(ωn)dx≥1p∫RNf¯(ωn)ωndx−∫RNF¯(ωn)dx≥∫RN[1pf¯(ω~)ω~−F¯(ω~)]dx+on(1).

So by (2.5) and (2.16), we have

0≤∫RN[1pf¯(ω~)ω~−F¯(ω~)]dx≤0,

which means ω̃ ≡ 0, contradicting to (2.18).

Thus, {u_n} is bounded in W^2,p(ℝ^N).

Next, we prove that J^∞ > 0. To this end, we take ρ_n = |u_n|^p. Following from Sobolev imbedding, we have that {ρ_n} is bounded in L¹(ℝ^N). Therefore, Lemma 2.1 implies that for some subsequence of {ρ_n} either Vanishing or Nonvanishing occurs. We will show that both Vanishing and Nonvanishing are impossible if J^∞ = 0.

Vanishing is impossible:

In fact, if Vanishing occurs, by Lemma 2.2 and (2.2), we have

I∞(un)=1p∥un∥p+on(1). (2.19)

Taking ε = m2 in (2.1), it follows from (2.11) that

∥un∥p=∫RNf¯(un)undx≤12∥un∥p+C∥un∥τ, for τ∈(p,p∗∗),

which means that there exists a δ > 0 such that

∥un∥≥δ. (2.20)

So, if I^∞(u_n) → J^∞ = 0 as n → +∞, (2.19) and (2.20) can deduce a contradiction.
Nonvanishing is also impossible:

In fact, if Nonvanishing occurs, there exist η > 0, R > 0, {y_n} ⊂ ℝ^N such that

limn→+∞∫BR(yn)|un|pdx≥η>0. (2.21)

Let ũ_n(x) = u_n(x + y_n). Since {u_n} is bounded in W^2,p(ℝ^N), {ũ_n} is also bounded in W^2,p(ℝ^N), then by Sobolev imbedding, we may assume that there exists 0 ≢ ũ ∈ W^2,p(ℝ^N), such that as n → +∞

u~n⇀u~ in W2,p(RN),u~n→u~ in Llocp(RN),u~n→u~ a.e. in RN. (2.22)

It is easy to see that

I∞(un)=I∞(u~n) and 〈I∞′(un),un〉=〈I∞′(u~n),u~n〉.

So by (2.5), (2.10)-(2.11) and Fatou’s Lemma, we have

0<∫RN[1pf¯(u~)u~−F¯(u~)]dx ≤lim infn→+∞∫RN[1pf¯(u~n)u~n−F¯(u~n)]dx=lim infn→+∞[1p∥u~n∥p−∫RNF¯(u~n)dx]=J∞=0,

which means that Nonvanishing is also impossible.

Above arguments show that both Vanishing and Nonvanishing are impossible if J^∞ = 0. This contradiction gives that J^∞ > 0.□

3 Proof of Theorem 1.2

This section will devote to the proof of Theorem 1.2.

Proof of Theorem 1.2

By Lemma 2.6, we have J^∞ > 0. By Lemma 2.5, there exists {u_n} ⊂ W^2,p(ℝ^N) such that as n → +∞

I∞(un)→J∞>0, (3.1)

I∞′(un)→0 in (W2,p(RN))−1, (3.2)

i.e., {u_n} is a (PS)_J^∞ sequence. Now we divide the proof into two steps.

{u_n} is bounded in W^2,p(ℝ^N).

If ∥u_n∥ → +∞, as n → +∞, we let

tn=α∥un∥, ωn(x)=tnun(x)=αun(x)∥un∥, (3.3)

where α is chosen such that α^p ≥ 2pJ^∞ > 0. Set ρ_n = |ω_n|^p, if there is a subsequence, still denoted by {ρ_n}, such that Lemma 2.1 holds, then by the similar arguments of (2.15)-(2.16), we know that Vanishing doesn’t occur.

If Nonvanishing occurs, i.e. there exist η > 0, R > 0 and {y_n} ⊂ ℝ^N such that

limn→+∞∫BR(yn)|ωn|pdx≥η>0,ω~n=ωn(x+yn), ∥ω~n∥=∥ωn∥=α.

Hence, there exists some 0 ≢ ω̃ ∈ W^2,p(ℝ^N) satisfying as n → +∞

ω~n⇀ω~ in W2,p(RN),ω~n→ω~ in Llocτ(RN),τ∈(p,p∗∗),ω~n→ω~ a.e. in RN. (3.4)

Set ũ_n(x) = u_n(x + y_n). Then ω̃_n = t_nũ_n, and it is not difficult to verity that ∀ϕ ∈ C0∞ (ℝ^N)

〈I∞′(u~n),ϕ〉=on(1), (3.5)

that is

∫RN(|Δu~n|p−2Δu~nΔϕ+m|u~n|p−2u~nϕ)dx−∫RNf¯(u~n)ϕdx=on(1).

Then, ∀ϕ ∈ C0∞ (ℝ^N)

∫RN(|Δω~n|p−2Δω~nΔϕ+m|ω~n|p−2ω~nϕ)dx−∫RNf¯(u~n)|u~n|p−2u~n|ω~n|p−2ω~nϕdx=on(1). (3.6)

Now we claim that as n → +∞

f¯(u~n)|u~n|p−2u~n|ω~n|p−2ω~n→l|ω~|p−2ω~a.e. x∈RN. (3.7)

Indeed, let

Ω1={x∈RN:ω~(x)≠0}andΩ2={x∈RN:ω~(x)=0}.

If x ∈ Ω₁, then we have |ũ_n(x)| = α^–1 ∥u_n(x)∥ ⋅ |ω̃_n(x)| → +∞, as n → +∞. Hence by (2.4), we have as n → +∞

f¯(u~n)|u~n|p−2u~n|ω~n|p−2ω~n→l|ω~|p−2ω~,a.e. x∈Ω1.

Since |f¯(t)||t|p−1≤l and ω̃_n(x) → ω̃(x) = 0 a.e. x ∈ Ω₂, it follows that as n → +∞

f¯(u~n)|u~n|p−2u~n|ω~n|p−2ω~n→0=l|ω~|p−2ω~,a.e. x∈Ω2.

Therefore, (3.7) is proved. Let ϕ ∈ C0∞ (ℝ^N) be arbitrary and fixed, and let Ω ⊂ ℝ^N be a compact set such that supp ϕ ⊂ Ω. The compactness of the Sobolev embedding W^2,p(Ω) ↪ L^p–1(Ω) implies ω̃_n → ω̃ strongly in L^p–1(Ω). Therefore, it follows from [24], by using the Lebesgue Dominated Theorem, then as n → +∞

∫RNf¯(u~n)|u~n|p−2u~n|ω~n|p−2ω~nϕdx→l∫RN|ω~|p−2ω~ϕdx. (3.8)

Similarly, for any ϕ ∈ C0∞ (ℝ^N), as n → +∞

∫RNm|ω~n|p−2ω~nϕdx→∫RNm|ω~|p−2ω~ϕdx. (3.9)

On the other hand, by (3.4)-(3.5), we have that

〈I∞′(ω~n)−I∞′(ω~),ηρ(ω~n−ω~)〉=on(1) (3.10)

for any cut-off function η_ρ ∈ C0∞ (ℝ^N) with 0 ≤ η_ρ ≤ 1, η_ρ = 1 on B_ρ(0) = {x ∈ ℝ^N : |x| ≤ ρ}. Next we are going to prove that as n → +∞

Δω~n→Δω~ a.e. in RN. (3.11)

Following the idea in [17, 25], we let

Pn(x)=(|Δω~n|p−2Δω~n−|Δω~|p−2Δω~)(Δω~n−Δω~).

By the well-known inequality

(|ξ|γ−2ξ−|η|γ−2η)(ξ−η)>0

for any γ > 1 and ξ, η ∈ ℝ^N with ξ ≠ η, we have P_n(x) ≥ 0. By (3.4) and (3.10), it is easy to see that

limn→+∞∫Bρ(0)Pn(x)dx=0for any ρ>0.

Then as n → +∞

Δω~n⇀Δω~ in Lp(Bρ(0)).

Since ρ > 0 is arbitrary, we know that (3.11) holds. Hence as n → +∞

|Δω~n|p−2Δω~n⇀|Δω~|p−2Δω~ in Lpp−1(RN).

So for any ϕ ∈ C0∞ (ℝ^N), as n → +∞

∫RN|Δω~n|p−2Δω~nΔϕdx→∫RN|Δω~|p−2Δω~Δϕdx. (3.12)

Combining (3.6), (3.8), (3.9) and (3.12), we have for any ϕ ∈ C0∞ (ℝ^N)

∫RN|Δω~|p−2Δω~Δϕdx=(l−m)∫RN|ω~|p−2ω~ϕdx,

which contradicts to Corollary 1.6, and Step 1 is completed.
J^∞ is achieved by some ũ ∈ W^2,p(ℝ^N) ∖ {0}.

Let ρ_n = |u_n|^p. Up to a subsequence, we may assume that {ρ_n} satisfies Lemma 2.1.

If Vanishing occurs, then it follows from Lemma 2.2 and (2.1)-(2.2) that

∫RNF¯(un)dx→0and∫RNf¯(un)undx→0as n→+∞.

Since

0=〈I∞′(un),un〉=∥un∥p−∫RNf¯(un)undx,

then as n → +∞

I∞(un) =1p∥un∥p−∫RNF¯(un)dx=∫RN[1pf¯(un)un−F¯(un)]dx→0,

which contradicts to (3.1).

So only Nonvanishing occurs. Similar to the arguments in Step 1, there exists 0 ≢ ũ ∈ W^2,p(ℝ^N) such that

∫RN(|Δu~|p−2Δu~Δϕ+m|u~|p−2u~ϕ)dx=∫RNf¯(u~)ϕdx, ∀ϕ∈C0∞(RN). (3.13)

Hence ũ ∈ Λ and I^∞(ũ) ≥ J^∞ > 0. On the other hand, since

0=〈I∞′(un),un〉=〈I∞′(u~n),u~n〉=∥u~n∥p−∫RNf¯(u~n)u~ndx,

then by (2.5) and Fatous’s Lemma, one has

J∞ =I∞(un)+on(1)=I∞(u~n)+on(1)=1p∥u~n∥p−∫RNF¯(u~n)dx+on(1)=∫RN[1pf¯(u~n)u~n−F¯(u~n)]dx+on(1)≥∫RN[1pf¯(u~)u~−F¯(u~)]dx+on(1)=I∞(u~)+on(1),

which implies that J^∞ ≥ I^∞(ũ).

So I^∞(ũ) = J^∞ with ũ ≢ 0.

For the case l ≤ m, we assume that Problem (1.6) has a nontrivial weak solution u ∈ W^2,p(ℝ^N). Then, following from Proposition 1.4, (2.3) and (2.5), we have

0 ≤N−2pNp∫RN|Δu|p dx=∫RN(F¯(u)−1pm|u|p) dx<1p∫RN(f¯(u)u−m|u|p) dx≤1p∫RN(l−m)|u|p dx≤0,

which is impossible. So when l ≤ m, there is no nontrivial weak solutions to (1.6).

We complete the proof.□

4 Proof of Theorem 1.3

We will put the proof of Theorem 1.3 into this section. First, we can verify that the functional I defined in (1.2) exhibits the Mountain Pass geometry.

Lemma 4.1

Assume that (C₁) – (C₃) and (C₅) hold. Then the functional I satisfies

there exist α₀ and ρ₀ > 0 such that I(u) ≥ α₀ for all ∥u∥ = ρ₀;
there exists w ∈ W^2,p(ℝ^N) such that ∥w∥ > ρ₀ and I(w) ≤ 0.

Proof

By (2.2), for any ε > 0 there exists a C_ε > 0 such that for some τ ∈ (p, p^**)

I(u) =1p∫RN(|Δu|p+m|u|p)dx−∫RNF(x,u)dx≥1p∫RN(|Δu|p+m|u|p)dx−∫RN(ε|u|p+Cε|u|τ)dx≥C∥u∥p−C∥u∥τ.

If we choose ∥u∥ = ρ₀ small, then I(u) ≥ α₀ > 0. We can get the result (i).

On the other hand, Let ũ be the ground state solution obtained in Theorem 1.2. We have that if x ∈ Ω₀ := {x ∈ ℝ^N : ũ(x) = 0}, then

F¯(tu~)tp=0=1pf¯(u~)u~,

and if x ∈ Ω := {x ∈ ℝ^N : ũ(x) ≠ 0}, then

lim inft→+∞F¯(tu~)tp=limt→+∞f¯(tu~)u~ptp−1=1p|u~|plimt→+∞f¯(tu~)|tu~|p−2tu~≥1p|u~|pf¯(u~)|u~|p−2u~=1pf¯(u~)u~.

Thus,

lim supt→+∞∫RN(1pf¯(u~)u~−F¯(tu~)tp)dx≤∫RN(1pf¯(u~)u~−liminft→+∞F¯(tu~)tp)dx≤0,

which implies that, for t large enough,

I∞(tu~) =1ptp∫RN(|Δu~|p+m|u~|p)dx−∫RNF¯(tu~)dx=1ptp∫RNf¯(u~)u~dx−∫RNF¯(tu~)dx=tp∫RN(1pf¯(u~)u~−F¯(tu~)tp)dx≤0. (4.1)

So there exists w = t₀ũ ∈ W^2,p(ℝ^N)(t₀ is large enough) such that

I∞(w)≤0 and ||w||>ρ0.

By (C₅), we have

I(w)≤I∞(w)≤0.

We complete the proof.□

As a consequence of Lemma 4.1 and the Mountain Pass Theorem without Cerami condition, founded in [26], for the constant

c=infγ∈Γmax0≤t≤1I(γ(t)), (4.2)

where

Γ={γ∈C([0,1],W2,p(RN)):γ(0)=0,γ(1)=w},

there exists a Cerami sequence {u_n} ⊂ W^2,p(ℝ^N) at the level c, that is

I(un)→cand(1+∥un∥)∥I′(un)∥(W2,p(RN))−1→0,as n→+∞. (4.3)

Lemma 4.2

The sequence {u_n} obtained in (4.3) is bounded.

Proof

Just replacing J^∞ by c, and following exactly the same procedures as in Step 1 in the proof of Theorem 1.2, we know that if ∥u_n∥ → +∞ as n → +∞, by Lemma 2.1, Vanishing can not happen. If Nonvanishing occurs, then there exist η > 0, R > 0 and {y_n} ⊂ ℝ^N such that

limn→+∞∫BR(yn)|ωn|pdx≥η>0.

Let ω̃_n(x) = ω_n(x + y_n), then ∥ω̃_n∥ = ∥ω_n∥ and there exists 0 ≢ ω̃ ∈ W^2,p(ℝ^N) such that as n → +∞

ω~n⇀ω~ in W2,p(RN),ω~n→ω~ in Llocτ(RN),τ∈(p,p∗∗),ω~n→ω~ a.e. in RN. (4.4)

Let ũ_n(x) = u_n(x + y_n), and for any ϕ ∈ C0∞ (ℝ^N), we have

〈I′(un),ϕ(x−yn)〉=on(1).

Hence

∫RN(|Δu~n|p−2Δu~nΔϕ+m|u~n|p−2u~nϕ)dx−∫RNf(x+yn,u~n)ϕdx=on(1),

that is, for any ϕ ∈ C0∞ (ℝ^N), we have

∫RN(|Δω~n|p−2Δω~nΔϕ+m|ω~n|p−2ω~nϕ)dx−∫RNf(x+yn,u~n)|u~n|p−2u~n|ω~n|p−2ω~nϕdx=on(1). (4.5)

Similar to the Nonvanishing case in Step 1 of Theorem 1.2, we can also prove that for any ϕ ∈ C0∞ (ℝ^N)

∫RN|Δω~|p−2Δω~Δϕdx=(l−m)∫RN|ω~|p−2ω~ϕdx,

which contradicts to Corollary 1.6, and the Lemma is proved.□

Lemma 4.3

Assume that the assumptions of Theorem 1.3 hold. Then c < J_∞.

Proof

Assume that w ∈ W^2,p(ℝ^N) is the function given in Lemma 4.1, we set γ̃(t) = tw. Then γ̃ ∈ Γ. Hence, by the definition of c and Lemma 2.3,

c≤maxt∈[0,1]I(γ~(t))≤maxt>0I(tu~(x))<maxt>0I∞(tu~(x))=I∞(u~(x))=J∞.

We complete the proof.□

Finally, we give the proof of Theorem 1.3.

The Proof of Theorem 1.3

Since {u_n} is bounded in W^2,p(ℝ^N), then there exist some u₀ ∈ W^2,p(ℝ^N) and a subsequence of {u_n}, still denoted by {u_n}, such that as n → +∞

un⇀u0 in W2,p(RN),un→u0 in Llocτ(RN),τ∈(p,p∗∗),un→u0 a.e. in RN. (4.6)

By (4.6) and similar to the proof of Step 1 in Theorem 1.2, we can prove that

Δun→Δu0 a.e. in RN, as n→+∞.

By (2.1)-(2.2), (4.3) and Lebesgue Dominated Theorem, we have that for any ϕ ∈ C0∞ (ℝ^N)

〈I′(un)−I′(u0),ϕ〉=∫RN(|Δun|p−2Δun−|Δu0|p−2Δu0)Δϕdx+∫RNm(|un|p−2un−|u0|p−2u0)ϕdx−∫RN[f(x,un)−f(x,u0)]ϕdx→0,asn→+∞.

Thus I′(u₀) = 0. In order to complete the proof of Theorem 1.3, we must show u₀ is nontrivial. To this end, we suppose by contradiction that u₀ ≡ 0.

u_n ↛ 0 in W^2,p(ℝ^N) as n → +∞.

Assume that u_n → 0 in W^2,p(ℝ^N) as n → +∞. Then limn→+∞ I(u_n) = 0, which contradicts to limn→+∞ I(u_n) = c > 0.
If u_n ⇀ 0 in W^2,p(ℝ^N) as n → +∞, then

limn→+∞[I(un)−I∞(un)]=0, limn→+∞[〈I′(un),un〉−〈I∞′(un),un〉]=0. (4.7)

In fact, for any given R > 0, we have that

|I(un)−I∞(un)| ≤∫RN|F(x,un)−F¯(un)|dx≤∫|x|≤R|F(x,un)−F¯(un)|dx+∫|x|≥R|F(x,un)−F¯(un)|dx≜In1+In2.

Since u_n ⇀ 0 in W^2,p(ℝ^N) as n → +∞, by Sobolev imbedding and up to a subsequence, we have that u_n → 0 in Llocτ (ℝ^N) as n → +∞, where τ ∈ [p, p^**). Hence (2.2) implies that limn→+∞In1=0.

On the other hand, by (C₂) – (C₃) and (C₅) for any δ > 0, we have

In2 =[∫{|x|≥R:|un|<δ}+∫{|x|≥R:δ≤|un|≤1δ}+∫{|x|≥R:|un|>1δ}]|F(x,un)−F¯(un)|dx≤ε1(δ)∫RN|un|pdx+ε2(R)δ−p∫RN|un|pdx+ε3(δ)∫RN|un|p∗∗dx,

where

ε1(δ)=sup{|x|≥R:|t|<δ}|F(x,t)−F¯(t)||t|p→0 as δ→0+,ε3(δ)=sup{|x|≥R:|t|>1δ}|F(x,t)−F¯(t)||t|p∗∗→0 as δ→0+andε2(R)=sup{|x|≥R:δ≤|t|≤1δ}|F(x,t)−F¯(t)||t|p→0 as R→+∞, for fixed δ.

Therefore, limn→+∞In2=0 and hence limn→+∞[I(un)−I∞(un)]=0. Similarly, we can deduce that limn→+∞ [〈I′(u_n), u_n〉 – 〈I^{∞^′}(u_n), u_n〉] = 0. So (4.7) is proved.
There is a A > 0 such that

∫RN|Δun|pdx≥A>0, ∀n≥1. (4.8)

In fact, if (4.8) is false, then ∫_ℝ^N|Δu_n|^pdx → 0 as n → +∞. Using Claim I, we may assume that for some η > 0

un∈W2,p(RN)∖{0} for all n≥1 and limn→+∞∥un∥=η,

which, together with ∫_ℝ^N|Δu_n|^pdx → 0 as n → +∞, implies that

(∫RNm|un|pdx)1p→η>0as n→+∞.

By (C₁) – (C₃), we have that for any ε > 0, there exists C_ε > 0 such that

f(x,un)un≤ε|un|p+Cε|un|p∗∗.

Then

on(1) =〈I′(un),un〉≥∫RN(|Δun|p+m|un|p)dx−∫RN(ε|un|p+Cε|un|p∗∗)dx≥∫RN(|Δun|p+m|un|p)dx−ε∫RN|un|pdx−C(∫RN|Δun|pdx)p∗∗p≥Cηp+on(1),

which is a contradiction. So (4.8) is true.

Based on the above three Claims, we would show that there is a contradiction, which implies that u₀ ≠ 0. By (4.3), (4.6) and (4.7), we have

c=I(un)+on(1)=I∞(un)+on(1),〈I∞′(un),un〉=on(1). (4.9)

Let ū_n(x) = u_n(t_nx), where t_n > 0 will be determined later. Then

I∞(u¯n)=1ptn−N(tn2p−1)∫RN|Δun|pdx+tn−NI∞(un) (4.10)

and

〈I∞′(u¯n),u¯n〉=tn−N[(tn2p−1)∫RN|Δun|pdx+〈I∞′(un),un〉]. (4.11)

Taking tn=|(1−〈I∞′(un),un〉∫RN|Δun|pdx)|12p, then by (4.8), (4.9) and (4.11) we know that

tn→1 as n→+∞, and u¯n∈Λ for n large enough. (4.12)

Combining (4.9)-(4.10) and (4.12), one has

c=I∞(un)+on(1)=I∞(u¯n)+on(1)≥J∞+on(1), (4.13)

which contradicts to Lemma 4.3.

Therefore u₀ ∈ W^2,p(ℝ^N) ∖ {0}.

Going on as the proof of Theorem 1.2, we can get the nonexistence of nontrivial weak solutions for the case l ≤ m. So the proof of Theorem 1.3 is complete.□

Acknowledgments

The authors would like to thank Professor Yinbin Deng for stimulating discussions and helpful suggestions. This paper was supported by the fund from NSF of China (No. 11701107, 11701108, 11926332, 11926320) and the NSF of Guangxi Province (2017GXNSFBA198190).

References

[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.10.1016/0022-1236(73)90051-7Search in Google Scholar

[2] G. B. Li and H. S. Zhou, Asymptotically linear Dirichlet problem for the p-Laplacian, Nonlinear Anal. 43 (2001), 1043–1055.10.1016/S0362-546X(99)00243-6Search in Google Scholar

[3] G. B. Li and H. S. Zhou, Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as u^p–1 at infinity, J. London Math. Soc. (2) 65 (2002), 123–138.10.1112/S0024610701002708Search in Google Scholar

[4] Y. Liu and Z. P. Wang, Biharmonic equations with asymptotically linear nonlinearities, Acta Math. Sci. Ser. B 27 (2007), 549–560.10.1016/S0252-9602(07)60055-1Search in Google Scholar

[5] H. S. Zhou, Existence of asymptotically linear Dirichlet problem, Nonlinear Anal. 44 (2001), 909–918.10.1016/S0362-546X(99)00314-4Search in Google Scholar

[6] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, (I), Arch. Ration. Mech. Anal. 82 (1983), 316–338.Search in Google Scholar

[7] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, (II), Arch. Ration. Mech. Anal. 82 (1983), 347–369.10.1007/BF00250556Search in Google Scholar

[8] W. A. Struass, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.10.1007/BF01626517Search in Google Scholar

[9] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I, Ann. Inst. H. Poincaré. Anal. Non Linéaire 1 (1984), 109–145.10.1016/s0294-1449(16)30428-0Search in Google Scholar

[10] P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré. Anal. Non Linéaire 1 (1984), 223–283.10.1016/s0294-1449(16)30422-xSearch in Google Scholar

[11] J. F. Yang and X. P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains, (I) positive mass case, Acta Math. Sci. 7 (1987), 341–359.10.1016/S0252-9602(18)30457-0Search in Google Scholar

[12] Y. B. Deng, Q. Gao and L. Y. Jin, On the existence of nontrivial solutions for p-harmonic equations on unbounded domains, Nonlinear Anal. 69 (2008), 4713–4731.10.1016/j.na.2007.11.024Search in Google Scholar

[13] D. G. Costa and O. H. Miyagaki, Nontrivial solutions for perturbations of p-Laplacian on unbounded domain, J. Math. Anal. Appl. 193 (1995), 737–755.10.1006/jmaa.1995.1264Search in Google Scholar

[14] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ℝ^N, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.10.1017/S0308210500013147Search in Google Scholar

[15] Y. Y. Li, Nonautonomous nonlinear scalar field equations, Indiana Univ. Math. J. 39 (1990), 283–301.10.1006/jdeq.1993.1058Search in Google Scholar

[16] C. A. Stuart and H. S. Zhou, A variational problems related to self-trapping of an eletromagnetic field, Math. Methods Appl. Sci. 19 (1996), 1397–1407.10.1002/(SICI)1099-1476(19961125)19:17<1397::AID-MMA833>3.0.CO;2-BSearch in Google Scholar

[17] C. A. Stuart and H. S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on ℝ^N, Comm. Partial Differential Equations 24 (1999), 1371–1385.10.1080/03605309908821481Search in Google Scholar

[18] G. B. Li and H. S. Zhou, The existence of a positive solution to asymptotically linear scalar field equations, Proc. Roy. Soc. Edinburgh. Sect. A 130 (2000), 81–105.10.1017/S0308210500000068Search in Google Scholar

[19] C. J. He and G. B. Li, The existence of a nontrivial solution to the p&q-Laplacian problem with nonlinearity asymptotic to u^p–1 at infinity in ℝ^N, Nonlinear Anal. 68 (2008), 1100–1119.10.1016/j.na.2006.12.008Search in Google Scholar

[20] Deng Yinbin and Gao Qi, Multi-solution of nonhomogeneous p-harmonic equation, J. Central China Normal University 39 (2005), no. 4, 444-447.Search in Google Scholar

[21] Tian Yi, Polysolution of nonhomogeneous p-harmonic equation with Navier boundary value condition, J. Huazhong Normal University 43 (2009), no. 2, 181-185.Search in Google Scholar

[22] J. Q. Liu and X. Q. Liu, On the eigenvalue problem for the p-Laplacian operator in ℝ^N, J. Math. Anal. Appl. 379 (2011), 861–869.10.1016/j.jmaa.2011.01.075Search in Google Scholar

[23] K. C. Chang, Critical Point Theory and its Applications, Shanghai Sci. & Tech Press (1986) (in Chinese).Search in Google Scholar

[24] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA , 1996.10.1007/978-1-4612-4146-1Search in Google Scholar

[25] G. B. Li and O. Martio, Stability in obstacle problems, Math. Scand. 75 (1994), 87–100.10.7146/math.scand.a-12505Search in Google Scholar

[26] I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer-Verlag, Berlin, 1990.10.1007/978-3-642-74331-3Search in Google Scholar

Received: 2020-06-23

Accepted: 2021-01-07

Published Online: 2021-03-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|^p–2t at infinity

Abstract

1 Introduction

Definition 1.1

Theorem 1.2

Theorem 1.3

Proposition 1.4

Remark 1.5

Corollary 1.6

2 Some notations and preliminary Lemmas

The proof of Proposition 1.4

Lemma 2.1

Lemma 2.2

Lemma 2.3

Proof

Lemma 2.4

Proof

Lemma 2.5

Lemma 2.6

Proof

3 Proof of Theorem 1.2

Proof of Theorem 1.2

4 Proof of Theorem 1.3

Lemma 4.1

Proof

Lemma 4.2

Proof

Lemma 4.3

Proof

The Proof of Theorem 1.3

Acknowledgments

References

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