Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to

where m > 0 is a constant, N > 2p ≥ 4 and lim t→∞ f (x,t) |t|p−2 t = 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 arti cial constraint method and the Pohozaev identity.


Introduction
In this paper, we deal with the existence and nonexistence of nontrivial solutions to the following p-harmonic problem    ∆(|∆u| p− ∆u) + m|u| p− u = f (x, u), x ∈ R N , where m > denotes a constant and N > p ≥ . Throughout this paper, we assume that f (x, t) satis es the following conditions: (C ) lim t→∞ f (x, t) |t| p− t = l uniformly in x ∈ R N for some l ∈ ( , +∞).
(C ) For a.e. x ∈ R N , f (x,t) |t| p− t is nondecreasing with respect to t > , and nonincreasing with respect to t < .
De nition 1.1. We call u ∈ W ,p (R N ) a (weak) solution of (1.1) if for all ϕ ∈ W ,p (R N ), we have It is easy to see that any solution of (1.1) corresponds to a critical point of the following energy functional de ned on W ,p (R N ): where F(x, u) = u 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 (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).
Deng, Gao and Jin [12] studied the p-harmonic problem (1.1). The authors in [12] required that f (x, t) is subcritical and satis es some assumptions similar to (C ) − (C ), (C ) − (C ) and the (AR) condition: This condition implies that for some C > , However, (C ) implies that f (x, t) is asymptotic to |t| p− t at in nity, 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 (C ) − (C ). 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 (C )−(C ). 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− t at in nity.
We rst de ne the limited problem of (1.1) as follows: (1.6) For any u ∈ W ,p (R N ), we de ne where c, C > and d = Np−p N− p . If u is a weak solution of the p-harmonic problem (1.6), that is 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 di culties: 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 ,p (R N ) implies that |u|, u + , u − ∈ W ,p (R N ), where u + = max{u, }, u − = max{−u, }. 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 ,p (R N ) does not imply that |u|, u + , u − ∈ W ,p (R N ). To this end, we need to take prolongation on f (x, t) for t < .
Secondly, as we can see in [18], the Pohozaev identity played an important role in the proofs of the main results, which is slightly di erent from what [19] did. However, we can not get the Pohozaev identity of pharmonic 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 N j= x j D h j u as a test function to derive the corresponding Pohozaev identity, which relaxs the restriction on the regularity of the solution and where D h j u denotes the di erence quotient and ψ is a given cut-o function. But our problem (1.6) is a quasilinear elliptic equation of fourth order or more, we need to estimate the higher order di erence. Thus more delicate analysis is needed.
Thirdly, W ,p (R N ) is not a Hilbert space in general. It is not clear that, up to a subsequence, Hence, more delicate analysis is needed to prove that ∆un → ∆u a.e. in R N .
Finally, since f (x, t) andf (t) are asymptotic to |t| p− t at in nity, 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 ,p (R N ) has become one of the main di culties for studying the existence of nontrivial weak solutions to (1.1) or (1.6) in W ,p (R N ). To show Theorem 1.2, inspired by [18], we apply Ekeland's variational principle to get a minimizing sequence {un} for J ∞ with I ∞′ (un) → in (W ,p (R N )) − , which guarantees that we can show J ∞ > . Then we can show that J ∞ is achieved by some u . 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 solutionũ to (1.6) obtained in Theorem 1.2, we construct the mountain pass level c as , W ,p (R N )) : γ( ) = , γ( ) = t ũ} for some t > large enough and which is slightly di erent from what [18] did. Liu and Zhou [18] useũ( x t ) instead of t ũ in the de nition of Γ and prove that I(ũ( x t )) < for t large enough. But for the solutions of (1.6), we can not obtain the regularity result to ensure that γ (t) = ũ( x tt ), t ∈ ( , ], , t = , belongs to Γ, which is a key point to show c < J ∞ . So the method in [18] does not work here. Due to lim |t| p− t = l, (C ) and (C ), we can show that I(tũ) < I ∞ (tũ) < for t > large enough, which implies that the mountain pass level c de ned above is well-de ned and c < J ∞ . By the mountain pass theorem without the Cerami condition, we can see that there exists a Cerami sequence {un} ⊂ W ,p (R N ) of I(u) at the level c, that is (1.12) Then we can apply the fact that c < J ∞ and the concentration-compactness principle to prove that {un} is bounded in W ,p (R N ) and the weak limit of such subsequence of {un} is a nontrivial weak solution of (1.1). This paper is organized as follows. In Section 2, we rst 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.

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 Let N > p and denote p ** = Np N− p . It follows from (C ) − (C ) that for any ε > , τ ∈ (p, p ** ), there exists According to (C ), ones can see that is strictly increasing in t > , and strictly decreasing in t < , Based on the above observations, we are going to prove Proposition 1.4.

The proof of Proposition 1.4:. Set
, where h ∈ R\{ } and e i denotes the unit vector along coordinate x i . We take a cut-o function ψ ∈ C ∞ (R N ) satisfying that ψ(x) = for |x| ≤ R, ψ(x) = for |x| ≥ R, |∇ψ| ≤ R and |∆ψ| ≤ R . Following the idea in [22], we set ϕ = ψ N j= x j D h j u, then we have ϕ ∈ W ,p (R N ). Taking such ϕ as a test function in (1.10), we have A straightforward computation gives us where D i u denotes ∇u · e i . Now we estimate the three terms I , I and I .
Since D h j u p ≤ C Du p , we have D h j u D j u weakly in L p (R N ), which means that, as h → , Since u satis es (1.10), we have . By the Hölder inequality, we have where p + q = . Additionally, where δ ij is the Kronecker symbol, that is δ ij = when i = j and otherwise. By Next we turn to estimate the right hand side of (2.6).
where we have used the facts that Altogether, as h → , by (2.6), we have Notice that |∇ψ| ≤ R , |∆ψ| ≤ R and supp ∇ψ, On the other hand, as R → +∞, Therefore we obtain the Pohozaev identity We list some useful Lemmas. |un| τ dx → for some R > , then un → in L α (R N ) as n → +∞, for any α ∈ (p, p ** ). 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).
Now we continue to study Λ. Byf (t) ∈ C (R), it is easy to see that the functional g(u) Then by (C ), for any u ∈ Λ g ′ (u), u = p So Λ is a closed complete submanifold of W ,p (R N ) with the natural Finsler structure (see [23]  Proof. If J ∞ = , then it follows from Lemma 2.5 that there exists a sequence {un} ⊂ Λ such that as n → +∞ Setωn(x) = ωn(x + yn), then ω n = ωn = α and by Sobolev imbedding, we may assume that for somẽ ω ∈ W ,p (R N ), such that as n → +∞ Letũn(x) = un(x+yn). Since {un} is bounded in W ,p (R N ), {ũn} is also bounded in W ,p (R N ), then by Sobolev imbedding, we may assume that there exists ≢ũ ∈ W ,p (R N ), such that as n → +∞ un →ũ in L p loc (R N ), un →ũ a.e. in R N .
So by (2.5), (2.10)-(2.11) and Fatou's Lemma, we have which means that Nonvanishing is also impossible. Above arguments show that both Vanishing and Nonvanishing are impossible if J ∞ = . This contradiction gives that J ∞ > .

Proof of Theorem 1.2
This section will devote to the proof of Theorem 1.2.
Step 1: {un} is bounded in W ,p (R N ).
Step  For the case l ≤ m, we assume that Problem (1.6) has a nontrivial weak solution u ∈ W ,p (R N ). Then, following from Proposition 1.4, (2.3) and (2.5), we have which is impossible. So when l ≤ m, there is no nontrivial weak solutions to (1.6).
We complete the proof.

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 de ned in (1.2) exhibits the Mountain Pass geometry. Proof. By (2.2), for any ε > there exists a Cε > such that for some τ ∈ (p, p ** ) If we choose u = ρ small, then I(u) ≥ α > . We can get the result (i).
On the other hand, Letũ be the ground state solution obtained in Theorem 1.2. We have that if which implies that, for t large enough, (4.1) So there exists w = t ũ ∈ W ,p (R N )(t is large enough) such that I ∞ (w) ≤ and ||w|| > ρ .
By (C ), we have We complete the proof.
We complete the proof.
Finally, we give the proof of Theorem 1.3.
The Proof of Theorem 1.3:. Since {un} is bounded in W ,p (R N ), then there exist some u ∈ W ,p (R N ) and a subsequence of {un}, still denoted by {un}, such that as n → +∞ a.e. in R N .
(4.6) By (4.6) and similar to the proof of Step 1 in Theorem 1.2, we can prove that ∆un → ∆u a.e. in R N , as n → +∞.
By (2.1)-(2.2), (4.3) and Lebesgue Dominated Theorem, we have that for any ϕ ∈ C ∞ (R N ) Thus I ′ (u ) = . In order to complete the proof of Theorem 1.3, we must show u is nontrivial. To this end, we suppose by contradiction that u ≡ . Claim I: un in W ,p (R N ) as n → +∞. Assume that un → in W ,p (R N ) as n → +∞. Then lim Since un in W ,p (R N ) as n → +∞, by Sobolev imbedding and up to a subsequence, we have that un → in L τ loc (R N ) as n → +∞, where τ ∈ [p, p ** ). Hence (2.2) implies that lim n→+∞ I n = .
On the other hand, by (C ) − (C ) and (C ) for any δ > , we have In fact, if (4.8) is false, then R N |∆un| p dx → as n → +∞. Using Claim I, we may assume that for some η > un ∈ W ,p (R N )\{ } for all n ≥ and lim n→+∞ un = η, which, together with R N |∆un| p dx → as n → +∞, implies that R N m|un| p dx p → η > as n → +∞.
Then on( ) = I ′ (un), un 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 u ≠ . By  Taking tn = | − I ∞′ (un),un R N |∆un| p dx | p , then by (4.8), (4.9) and (4.11) we know that tn → as n → +∞, andūn ∈ Λ for n large enough.