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

1 Introduction

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

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

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

1. 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.

2. limt→0f(x,t)|t|p−2t=0 uniformly in x ∈ ℝ^N.

3. limt→∞f(x,t)|t|p−2t=l uniformly in x ∈ ℝ^N for some l ∈ (0, +∞).

4. 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.

5. ∃ 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.

6. 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.

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):

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

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

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:

This condition implies that for some C > 0,

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:

For any u ∈ W^2,p(ℝ^N), we define

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

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

is well-defined.

Our main results can be stated as follows:

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

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

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,

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

Hence, more delicate analysis is needed to prove that

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

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

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

and the norm of u ∈ W^2,p(ℝ^N) by

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,

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

According to (C₆), ones can see that

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