Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

∂ Δy i N 1 1 yi 2 2 . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in N .

Abstract: In this paper, we study the following generalized Kadomtsev-Petviashvili equation in .
x xxx x x y N 1

(1.2)
If we choose ( ) = h s s 2 in (1.1), then equation (1.1) is a two-dimensional generalization of the Kortewegde Vries equation, which describes long dispersive waves in mathematical models, see [1]. When ( ) = | | h s s s p with = p m n , where m and n are relative prime numbers, and n is odd, Bouard and Saut [2,3] proved that there is a solitary wave for (1.1) with ≤ < p 1 4, if = N 2, or ≤ < p 1 pactness principle from [4,5]. In [6], Willem proved the existence of solitary waves of (1.1) as = N 2 and ∈ ( ) h C , 1 . In [7], Xuan extended the results obtained by [6] to higher dimension. In [8], ( ) h u was replaced by ( )| | − Q x y u u , p 2 , and Liang and Su had obtained nontrivial solutions of (1.1). In [9], Xu and Wei studied infinitely many solutions for with the Ambrosetti-Rabinowitz condition in bounded domain. For related contributions to study of solitary waves of the generalized Kadomtsev-Petviashvili equations, we refer to previous studies [10,11].
The aim of this paper is to prove the existence of multiple solutions of (1.3) in bounded domain without condition (AR), which is to ensure the boundedness of the (PS) sequences of the corresponding functional, and obtain the ground state solutions of (1.2) in N . In what follows, we assume that the function → h : satisfies the following conditions: Consider the following system, Now, we can state our first result. Our second result is as follows. Notations. Throughout the paper, we denote by ∥⋅∥ p the usual norm of Lebesgue space ( ) L p N . * X is the dual space of X. The symbol C denotes a positive constant and may vary from line to line.

Preliminary
In this section, we want to introduce the functional setting and some main results. At first, we present the functional setting (see [7,11]).
x x x y x y 1 1 N and the norm is , then we say that → u : N belongs to X.
, then we say that → u : N belongs to X 0 .
Then I possesses an unbounded sequence of critical values.
X, be a Banach space and ⊂ + T be an interval. Consider a family of C 1 functionals on X of the form Then, for almost every ∈ λ T , there exists a bounded ( ) PS cλ sequence in X, and the mapping → λ c λ is nonincreasing and left continuous.

Proof of Theorem 1.1
In this section, we consider the boundary value problem (1.3). The energy functional → I X : given by Proof. To prove that I satisfies the (PS) condition, we only need to prove { } ⊂ u X n has a convergent subsequence, where { } u n obtained by Lemma 3.1. As { } u n is bounded in X, there exists a subsequence still denoted by { } u n and ∈ u X 0 such that ⇀ u u n 0 in X and → u u Proof of Theorem 1.1. We have verified that I satisfies the (PS) condition. It follows from ( ) h 5 that I is an even function. As X is a separable space, X has orthonormal basis { } Next, we verify that I satisfies (ii) in Lemmas 2.2. By Lemma 2.1, for all ∈ u V, we have (ii) By virtue of ( ) h 2 , for any > ε 0 and some ∈ ( By Lemma 2.3, we have Then, there exists > ρ 0 small enough such that