Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2

. In this article, we study the generalized Kadomtsev-Petviashvili equation with a potential


Introduction
This article is devoted to studying solitary waves for the generalized Kadomtsev-Petviashvili equation with a potential where v = v(t, x, y) with (t, x, y) ∈ R + ×R×R, V is a potential function, parameter ε > 0 and D −1 x h(x, y) = x −∞ h(s, y)ds. A solitary wave is a solution of the form u(x−ct, y), with c > 0. Hence, inserting this into (1.1), we obtain x u yy in R 2 , or equivalently, where V = V + c, and by a simple scaling calculus, it is easy to see that the above equation becomes (1.2) (A5) There exist constants q, σ ∈ (2, 6), C 0 > 0 such that f (t) ≥ C 0 t q−1 , for all t ≥ 0, and lim t→∞ f (t) t σ−1 = 0. (A6) There is θ ∈ (2, 6) such that 0 < θF (t) = θ t 0 f (r)dr ≤ tf (t) for all t > 0.
(A7) f (t) > 0 for all t > 0, and f (t) = 0 for all t < 0. (A8) The function t → f (t)/t is increasing for t > 0. By using the variational method and the concentration-compactness principle, the authors obtained that the number of solitary waves corresponds to the number of global minimum points of the potential V when positive parameter ε is small enough. Notice that different from the works in [3,19], the use of special condition (A3) further ensures that the exact number of solutions can be secured in [10].
Motivated by the ideas developed in [1,13,18], we study problem (1.2) by considering a local assumption on V , the penalization scheme, and the Ljusternik-Schnirelmann theory. We aim to investigate the existence of multiple solutions for problem (1.2) without assuming (A2), (A3), (A6), and (A8). Furthermore, we assume additionally that V ∈ C(R 2 , R) f ∈ C 1 (R 2 , R) satisfy the following conditions: (A9) There exists an open and bounded set Ω ⊂ R 2 satisfying V 0 < min ∂Ω V and M = {x ∈ Ω : V (x) = V 0 } = ∅. (A10) There exists a positive number α ∈ (1, +∞) such that t → f (t) t α is nondecreasing on (0, ∞). Throughout this article, without loss of generality, we assume that V (0, 0) = V 0 = min x∈R 2 V (x). A typical example of function f which satisfying assumptions (A4), (A5), (A7), (A10) is Comparing this article with [10], we not only improve the above conditions of V and f , but also adopt different methods from [10]. In the proof of Theorem 1.1, we adopt the penalization method to restore the modified functional compactness. But in our equation (1.2), we notice that the local condition of potential V make the modified problem more complicated, thus we use the truncation trick from [21] to overcome this difficulties. It consists in making a suitable modification on the nonlinearity f , solving a modified problem and then check that for ε small enough, the solutions of the modified problem are indeed solutions of the original one. Moreover, we apply the method introduced by Benci and Cerami [4] to describe the multiplicity result. By considering the relationship between the category of some sublevel sets of the modified functional and the category of set M , we prove the existence of multiple solutions for the modified problem. Specifically, the main ingredient is to make precisely comparisons between the category of some sublevel sets of the modified functional and the category of the set M given in (A9). Remarkably, unlike the method in [1], we don't analyze the regularity and concentration of solutions, and we obtain the existence of estimates involving the L ∞ -norm of the modified problem by using Moser's iteration method [11]. To this end, we believe that the idea of combining penalization scheme with topological arguments to get the multiple solutions can be widely applied in different equations or systems to cope with local conditions on the potential V . Finally, by Chang's definition of category in [9], we remark that the category cat X (A) of a subset A of a topological space X is defined as the minimal k ∈ N such that A is covered by k closed subsets of X which are contractible in X, namely cat X (A) = inf k ∈ N ∪ {+∞} : ∃k contractible closed subsets of Our main results can be stated as follows.
The article is organized as follows. In Section 2, we give the variational setting and we modify the original problem. In Section 3, we study the autonomous problem associated with the modified problem. From this study, we obtain that the modified problem has multiple solutions by means of the Ljusternik-Schnirelmann theory. In Section 4, for ε > 0 small enough, we prove that the solutions of the modified problem are indeed solutions of the original problem by using Moser iteration method.
Throughout the article, we use the following notation: • · p denotes the norm of the Lebesgue space L p (R 2 ).

Variational framework
Arguing as in [25], in the set Y = {g x : g ∈ C ∞ 0 (R 2 )}, we define the inner product and the corresponding norm Now we define a working space of functions. A function u : R 2 → R belongs to X ε if there exists {u n } ⊂ Y such that (i) u n → u a.e. in R 2 ; (ii) u j − u k ε → 0 as j, k → ∞.
The space X ε with inner product (2.1) and norm (2.2) is a Hilbert space. By a solution of (1.2) we mean a function u ∈ X ε such that we define the Euler-Lagrange functional associated with (1.2) by The embedding X ε → L p (R 2 ) is continuous for 2 ≤ p ≤ 6 and X ε → L p loc (R 2 ) is compact for 2 ≤ p < 6 (see [8,5]).
As in [21], to study problem (1.2) by variational methods, we modify suitably the nonlinearity f so that, for parameter ε > 0 small enough, the solutions of the modified problem are also solutions of the original problem (1.2). To establish the multiplicity of solutions of (1.2), we will adapt for our case an argument explored by the penalization method introduced by Del and Felmer [21]. To this end, we need to fix some notation.
Let k > 2 and a > 0 such that f (a) = V 0 a/k with V 0 given by (A1). We set Using the functions ω and f , let us consider two new functions where χ(x, y) ∈ C ∞ 0 (R 2 ) is the characteristic function of set Ω, and g : k t for all t ≥ 0. Hereafter, for the above δ > 0, we have Now, we illustrate the properties of g. First, it follows from (A4) and (A10) that In view of (A4), (A5), (A7), and (A10), it is easy to deduce that g is a Carathéodory function and satisfying the following properties: The energy functional associated with problem (2.10) is It is standard to prove that ϕ ε ∈ C 1 (X ε , R) and its critical points are the weak solutions of the modified problem (2.10). Next, we define the Nehari manifold associated with ϕ ε given by The first lemma is related to the fact that ϕ ε satisfies the mountain pass geometry(see [25]).
Lemma 2.1. The functional ϕ ε satisfies the following properties.

Now by the Sobolev continuous embedding
Hence, we can choose some r, ρ > 0 such that ϕ ε (u) ≥ ρ with u ε = r small enough.
Therefore, t ε w ∈ N ε . Then we will prove the uniqueness of t ε , and we have Hence, we define a function q : By calculation and condition (A14), we prove that the function q is increasing on (0, +∞). Therefore, t ε is unique. The conclusion is obvious and the proof is complete.
Lemmas 2.1 and 2.3 permit us to apply the Mountain Pass Lemma due to Ambrosetti and Rabinowitz [25] to conclude that c ε is a critical value for ϕ ε .
Since {w n } is bounded, up to a subsequence, we may suppose that Φ ε (w n ), w n → a ≤ 0.

Multiplicity of solutions for the modified problem
3.1. Autonomous problem. Along all the section we shall assume that δ > 0 is small enough such that M δ ⊂ Ω, where Ω is given in the condition (A9). We start by considering the limit problem associated with (2.10), namely, the problem The solutions of (3.1) are precisely critical points of the functional defined by for all u, v ∈ X 0 . Let N 0 be the Nehari manifold associated with ϕ 0 by where X 0 is defined as the Hilbert space X ε but endowed with the inner product (3.5) and the corresponding norm As in the previous section, the next lemma characterizes of the infimum of ϕ 0 over N 0 . EJDE-2023/48 Lemma 3.1. If the conditions (A1), (A4), (A5), (A9) hold, then, for each u ∈ X 0 with u = 0, there exists a unique t 0 = t 0 (u) > 0 such that t 0 u ∈ N 0 and ϕ 0 (t 0 u) = max t≥0 ϕ 0 (tu). Moreover, we have where c V0 is the minimax level of Mountain Pass Theorem applied to ϕ 0 , namely where Γ = {η ∈ C([0, 1], X 0 ) : η(0) = 0 and ϕ 0 (η(1)) < 0}.
The proof of the above lemma is similar to the proof of Lemma 2.2. We omit it. The next lemma allows us to assume that the weak limit of a (P S) c V 0 sequence is nontrivial. Then w n → 0 strongly in L q (R 2 ), for every 2 < q < 6.
Proof. First of all, we fix q ∈ (2, 6). Given R > 0 and z ∈ R 2 , by standard interpolation inequality and Sobolev embedding theorem, we obtain , where 1−λ 2 + λ 6 = 1 q . Now, covering R 2 with balls of radius R, in such a way that each point of R 2 is contained in at most 3 balls, we find that Under the assumption of the lemma, we have w n → 0 in L q (R 2 ).
Therefore, there exists C 4 > 0 large enough such that On the other hand, by conditions (A10) and (2.9), we have The above inequality proves that {w n } is bounded in X 0 . Proof. Let {w n } ⊂ X 0 be a (P S) c V 0 sequence for ϕ 0 . By Lemma 3.4, we know that {w n } is bounded in X 0 . Then, up to a subsequence, w n w weakly in X 0 , w n → w strongly in L p loc (R 2 ), p ∈ [2, 6) and w n → w a.e. in R 2 . As in the proof of Lemma 2.1, we can easily prove that ϕ 0 satisfies the Mountain Pass Geometry. By the Mountain Pass Lemma (see [25]), there exists a Palais-Smale sequence {w n } for ϕ 0 at the mountain pass level c V0 . Moreover, w n w in X 0 and w is a critical point of ϕ 0 . From Lemma 3.2, we know that w = 0 and w ∈ N 0 .
Next we prove that w n → w strongly in X 0 . From the semi-lower continuity of norm, we have lim n→∞ w n 0 ≥ w 0 . (3.9) Observe that we must have the above equality hold. Otherwise, by Fatou's Lemma we obtain which is a contradiction. Thus, we conclude that up to a subsequence, So, the Brezis-Lieb Lemma [6] implies that w n → w in X 0 . Last, we prove that the solution w is nonnegative. From (A7) and using −w − as a testing function, we have where w − = max{−w, 0}. This implies that w ≥ 0 in R 2 is a nonnegative weak solution of (3.1) and the proof is complete.
3.2. Multiplicity of solutions for (2.10). In this subsection we will relate the number of solutions of (2.10) to the topology of the set M . For this, the next compactness result is fundamental for showing that the solutions of the modified problem are solutions of the original problem.  1]). Let ε n → 0 and {w n } ⊂ N εn be such that ϕ εn (w n ) → c V0 . Then there exists a sequence {z n } ⊂ R 2 such that v n (x) = w n (x +z n ) has a convergent subsequence in X 0 . Moreover, up to a subsequence, z n := ε nzn → z 0 ∈ M .

Proof of Theorem 1.1
In this section we shall prove our main result. The idea is to show that the solutions obtained in Theorems 3.5 and 3.9 satisfy the estimate u ε ≤ a, ∀x ∈ R 2 \ Ω for ε small enough. This fact implies that these solutions are indeed solutions of the original problem (1.2). The following lemma plays a fundamental role in the study of behavior of the maximum points of the solutions. Then, up to a subsequence, there exists a sequence (x n , y n ) ⊂ R 2 such that ψ n (x, y) := w n (x n + x, y n + y) satisfies that w n (x, y) = ψ n (x, y) ∈ L ∞ (R 2 ), and there exists C 5 > 0 such that w n L ∞ (R 2 ) ≤ C 5 , for all n ∈ N, (4.1) where (x n , y n ) =z n is given in Proposition 3.7.
The proof of the above lemma is similar to the proof of [13, Lemma 4.1]. We omit it.
Proof of Theorem 1.1. At first, we fix a number δ > 0 small such that M δ ⊂ Ω. Similar to the proof of [13, Theorem 1.1], there exists ε δ > 0 such that for any ε ∈ (0, ε δ ) and b > 0, then for any solution w ε ∈ N c V 0 +h(ε) ε of problem (2.10), it holds w ε L ∞ (R 2 \Ωε) < b. Let ε δ be given in Theorem 3.9 and ε := min{ ε δ , ε δ }. From Theorem 3.9, we can know that there exists cat M δ (M ) nontrivial solutions of problem (2.10) in N c V 0 +h(ε) ε . If w ∈ X ε is one of these solutions, then w ∈ N c V 0 +h(ε) ε , it follows from (4.2) and the definition of g that g(·, w) = f (w). Thus w is also a solution of the original problem (1.2). Then (1.2) has at least cat M δ (M ) nontrivial solutions. The proof of Theorem 1.1 is complete.
Jing Chen (corresponding author) School of Mathematics and Computer science, Hunan University of Science and Technology, Xiangtan, 411201 Hunan, China Email address: cjhnust@hnust.edu.cn