Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 48, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.48 MULTIPLICITY OF SOLUTIONS FOR A GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION WITH POTENTIAL IN R2 ZHENG XIE, JING CHEN Abstract. In this article, we study the generalized Kadomtsev-Petviashvili equation with a potential (−uxx + D−2 x uyy + V (εx, εy)u− f(u))x = 0 in R2, where D−2 x h(x, y) = ∫ x −∞ ∫ t −∞ h(s, y) ds dt, f is a nonlinearity, ε is a small positive parameter, and the potential V satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by apply- ing penalization techniques. The relationship between the number of positive solutions and the topology of the set where V attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser’s iteration method, we verify that the solutions of the modified problem are indeed solu- tions of the original problem for ε > 0 small enough. 1. Introduction This article is devoted to studying solitary waves for the generalized Kadomtsev- Petviashvili equation with a potential vt + ε2vxxx + (f(v)− Ṽ (x, y)v)x = D−1 x vyy, (1.1) where v = v(t, x, y) with (t, x, y) ∈ R+×R×R, Ṽ 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 −cux + ε2uxxx + (f(u)− Ṽ (x, y)u)x = D−1 x uyy in R2, or equivalently,( −ε2uxx +D−2 x uyy + V (x, y)u− f(u) ) x = 0, in R2, where V = Ṽ + c, and by a simple scaling calculus, it is easy to see that the above equation becomes( −uxx +D−2 x uyy + V (εx, εy)u− f(u) ) x = 0 in R2. (1.2) 2020 Mathematics Subject Classification. 35A15, 35A18, 35Q53, 58E05, 76B25. Key words and phrases. Kadomtsev-Petviashvili equation; variational methods; penalization techniques; Ljusternik-Schnirelmann theory. ©2023. This work is licensed under a CC BY 4.0 license. Submitted May 4, 2023. Published July 17, 2023. 1 2 Z. XIE, J. CHEN EJDE-2023/48 Let τ > 0 be a constant, if V (εx, εx) = τ , then problem (1.2) becomes( −uxx +D−2 x uyy + τu− f(u) ) x = 0 in R2. (1.3) If we choose f(t) = t2 in (1.3), then above equation becomes a well-known two- dimensional Korteweg-de Vries type equation, which models long dispersive waves, essentially unidimensional, but having small transverse effects, see [14]. In the pioneering work, De Bouard and Saut [7, 8] used the concentration compactness principle from [15, 16] to show the existence of solitary waves for (1.3) in the plane, with f(t) = |t|pt and p = m n , where m and n are relatively prime numbers and n is odd. Willem [25] extended the above results to the case N = 2 and f ∈ C(R,R) via the mountain pass theorem. Thereafter, Wang and Willem [26] proved multiple solutions for problem (1.3) by the Lyusternik-Schnirelman category theory. Liang and Su [17] obtained the existence of solutions for equation (1.3) with f(x, y, u) = Q(x, y)|u|p−2u and N ≥ 2, where Q ∈ C(R×RN−1,R), satisfying some structural conditions and 2 < p < p̄ = 2(2N−1) 2N−3 . Combining the variational method and linking theorems were appeared in [24], He and Zou [12] studied the existence of nontrivial solutions for the above equation in multi-dimensional spaces. Similar techniques were used in [30]. For further works about nontrivial solitary waves for the generalized Kadomtsev-Petviashvili equation, we refer to [2, 22, 27, 28, 29, 31] and the references therein. Recently, Alves and Miyagaki [3] obtained the existence, regularity and concen- tration phenomenon of nontrivial solitary waves for a class of generalized variable coefficient Kadomtsev-Petviashvili equation in R2. Lately, inspired by the simi- lar method of researching concentration in [3], Li, Wei and Yang [19] also studied concentration of solitary waves for problem (1.2) in R2 under the assumption of f(u) = |u|p−2u, where 2 < p < 6 and potential V satisfies a global condition. With the help of variational methods, the authors obtained that the existence of the least energy solution for all ε > 0 small enough and these solutions concentrate to the minimum point of the potential V as ε→ 0. We notice that potentials V in [3, 19] satisfy global conditions. However, the problem (1.2) becomes more complicated when potential V satisfies the local condi- tion. Alves and Ji [1] proved the existence and concentration of nontrivial solitary waves for (1.2) in R2 by using penalization method in [21]. Looking at their con- ditions, potential V and nonlinearity f required that V ∈ C2(R2,R) ∩ L∞(R2,R) and f : R→ R is C2 with |f ′′(t)| ≤ C|t|p−1 for all t ∈ R, which are so strong to ob- tain more regularity. Furthermore, by analyzing the regularity of the solutions, the authors obtained the concentration and estimation of the solutions. To the best of our knowledge, these analyses are crucial to prove that the solutions of the modified problem are indeed solutions of the original problem for ε > 0 small enough. As we know, there are only a few works about the multiple solutions for problem (1.2), Figueiredo and Montenegro [10] investigated the multiple solitary waves for problem (1.2). In their work, they assumed that both global potential V and nonlinearity f belong to C1 and the following conditions hold: (A1) There is a constant V0 > 0 such that V0 := infx∈R2 V (x). (A2) V∞ = lim|(x,y)|→∞ V (x, y) > V0. (A3) V −1({V0}) = {a1, a2, . . . , am} ⊂ R2 with a1 = 0 and aj 6= as if j 6= s. (A4) f ∈ C1(R,R) with f(0) = f ′(0) = 0. EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 3 (A5) There exist constants q, σ ∈ (2, 6), C0 > 0 such that f(t) ≥ C0t q−1, for all t ≥ 0, and lim t→∞ f(t) tσ−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 7→ 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(R2,R) f ∈ C1(R2,R) satisfy the following conditions: (A9) There exists an open and bounded set Ω ⊂ R2 satisfying V0 < min∂Ω V and M = {x ∈ Ω : V (x) = V0} 6= ∅. (A10) There exists a positive number α ∈ (1,+∞) such that t 7→ f(t) tα is nondecreasing on (0,∞). Throughout this article, without loss of generality, we assume that V (0, 0) = V0 = minx∈R2 V (x). A typical example of function f which satisfying assumptions (A4), (A5), (A7), (A10) is f(t) = { λtq−1 with λ > 0, 2 < q < 6, for t > 0, 0 for t ≤ 0 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 4 Z. XIE, J. CHEN EJDE-2023/48 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 catX(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 catX(A) = inf { k ∈ N ∪ {+∞} : ∃k contractible closed subsets of X : F1, F2, . . . , Fk such that A ⊂ ∪ki=1Fi } . where set F is called contractible (in X) if there exists κ : [0, 1]×X 7→ X such that κ(0, F ) = idX and κ(1, F ) is a one-point set. Our main results can be stated as follows. Theorem 1.1. Suppose that the conditions (A1), (A4), (A5), (A7), (A9), (A10) hold. Then, for any δ > 0 such that Mδ = {x ∈ R2 : dist(x,M) ≤ δ} ⊂ Ω, there exists ε̂ > 0 such that, for any ε ∈ (0, ε̂), problem (1.2) has at least catMδ (M) solutions. 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 prob- lem 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 mod- ified 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 Lp(R2). • for x ∈ R2 and r > 0, Br(x) := {y ∈ R2 : |y − x| < r}. • C1, C2, C3, . . . denote positive constants possibly different in different places. • u± = max{±u, 0}. • The symbols “→” and “⇀” denote strong and weak convergence, respec- tively. 2. Variational framework Arguing as in [25], in the set Y = {gx : g ∈ C∞0 (R2)}, we define the inner product (u, v)ε = ∫ R2 [uxvx +D−1 x uyD −1 x vy + V (εx, εy)uv] dx dy, (2.1) and the corresponding norm ‖u‖ε = (∫ R2 [|ux|2 + |D−1 x uy|2 + V (εx, εy)u2] dx dy )1/2 . (2.2) Now we define a working space of functions. A function u : R2 → R belongs to Xε if there exists {un} ⊂ Y such that (i) un → u a.e. in R2; (ii) ‖uj − uk‖ε → 0 as j, k →∞. EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 5 The space Xε with inner product (2.1) and norm (2.2) is a Hilbert space. By a solution of (1.2) we mean a function u ∈ Xε such that (u, φ)ε − ∫ R2 f(u)φdx dy = 0 for all φ ∈ Xε, we define the Euler-Lagrange functional associated with (1.2) by Iε(u) = 1 2 ∫ R2 [|ux|2 + |D−1 x uy|2 + V (εx, εy)u2] dx dy − ∫ R2 F (u) dx dy = 1 2 ‖u‖2ε − ∫ R2 F (u) dx dy, (2.3) which is C1 with Gateaux derivative 〈I ′ε(u), v〉 = ∫ R2 [uxvx +D−1 x uyD −1 x vy + V (εx, εy)uv] dx dy − ∫ R2 f(u)v dx dy = (u, v)ε − ∫ R2 f(u)v dx dy, ∀u, v ∈ Xε. (2.4) The embedding Xε ↪→ Lp(R2) is continuous for 2 ≤ p ≤ 6 and Xε ↪→ Lploc(R2) 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) = V0a/k with V0 given by (A1). We set f̂(t) = { f(t), i ft ≤ a, V0 k t, if t ≤ a. Moreover, we fix t0, t1 ∈ (0,+∞) such that t0 < a < t1 and ω ∈ C1([t0, t1]), satisfying ω(t) ≤ f̂(t) for all t ∈ [t0, t1], (2.5) ω(t0) = f̂(t0), ω(t1) = f̂(t1), (2.6) ω′(t0) = f̂ ′(t0), ω′(t1) = f̂ ′(t1), (2.7) t 7→ ω(t) t is nondecreasing for t ∈ [t0, t1]. (2.8) Using the functions ω and f̂ , let us consider two new functions f̃(t) = { f̂(t), if t /∈ [t0, t1], ω(t), if t ∈ [t0, t1]. and g(x, y, t) = χ(x, y)f(t) + (1− χ(x, y))f̃(t), where χ(x, y) ∈ C∞0 (R2) is the characteristic function of set Ω, and g : R2×R→ R is the penalized nonlinearity. 6 Z. XIE, J. CHEN EJDE-2023/48 From the definition of f̃ , it follows that f̃ ∈ C1(R,R), f̃(t) ≤ V0 k t for all t ≥ 0. Hereafter, for the above δ > 0, we have V0 + δ < min Ω̄δ V (x, y) where Ω̄δ = {x ∈ R2 : dist(x, Ω̄) ≤ δ}. Now, we illustrate the properties of g. First, it follows from (A4) and (A10) that 0 < 2F (t) < (α+ 1)F (t) ≤ f(t)t for all t > 0. (2.9) In view of (A4), (A5), (A7), and (A10), it is easy to deduce that g is a Carathéodory function and satisfying the following properties: (A11) g(x, y, t) ≤ δt+ f(t) for any t ≥ 0 and δ ≥ 0; (A12) limt→0 g(x,y,t) t = 0 uniformly in (x, y) ∈ R2; (A13) 0 < 2G(x, y, t) = 2 ∫ t 0 g(x, y, r)dr < g(x, y, t)t ≤ V0t 2/k for all (x, y) ∈ R2 \ Ω and all t > 0; (A14) For each (x, y) ∈ Ω, α > 1, the function t 7→ g(x,y,t) tα is increasing on (0,∞), and for each (x, y) ∈ R2 \Ω, the function t 7→ g(x,y,t) tα is increasing on (0, a). Now we study the modified problem( −uxx +D−2 x uyy + V (εx, εx)u− g(εx, εx, u) ) x = 0 in R2. (2.10) Notice that solutions of (2.10) with |u(x, y)| ≤ a for each (x, y) ∈ R2 \ Ωε are also the solutions of (1.2), where Ωε = {(x, y) ∈ R2 : ε(x, y) ∈ Ω}. The energy functional associated with problem (2.10) is ϕε(u) = 1 2 ‖u‖2ε − ∫ R2 G(εx, εy, u) dx dy. (2.11) It is standard to prove that ϕε ∈ C1(Xε,R) and its critical points are the weak solutions of the modified problem (2.10). Next, we define the Nehari manifold associated with ϕε given by Nε = {u ∈ Xε \ {0} : 〈ϕ′ε(u), u〉 = 0}. (2.12) The first lemma is related to the fact that ϕε satisfies the mountain pass geome- try(see [25]). Lemma 2.1. The functional ϕε satisfies the following properties. (i) There exist r, ρ > 0 such that ϕε(u) ≥ ρ with ‖u‖ε = r. (ii) There exists ‖e‖ε > r satisfying ϕε(e) < 0. Proof. (i) For any u ∈ Xε \ {0} and δ > 0 small, it follows from (A5), (A11), and (A12) that there exists Cδ > such that |g(εx, εy, t)| ≤ δ|t|+ Cδ|t|σ−1, for all (x, y) ∈ R2, t ∈ R, |G(εx, εy, t)| ≤ δ 2 t2 + Cδ σ |t|σ, for all (x, y) ∈ R2, t ∈ R. Now by the Sobolev continuous embedding Xε ↪→ Lp(R2) for p ∈ [2, 6], we have ϕε(u) = 1 2 ∫ R2 ( |ux|2 + |D−1 x uy|2 + V (εx, εy)u2 ) dx dy − ∫ R2 G(εx, εy, u) dx dy ≥ 1 2 ‖u‖2ε − δ 2 ∫ R2 u2 dx dy − Cδ σ ∫ R2 uσ dx dy EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 7 ≥ 1 + C1 4 ‖u‖2ε. Hence, we can choose some r, ρ > 0 such that ϕε(u) ≥ ρ with ‖u‖ε = r small enough. (ii) For each u ∈ Xε \ {0} with supp(u) ⊂ Ωε, and t > 0, we obtain that ϕε(tu) = t2 2 ‖u‖2ε − ∫ R2 G(εx, εy, tu) dx dy = t2 2 ‖u‖2ε − ∫ Ω̄ε F (tu) dx dy ≤ t2 2 ‖u‖2ε − t(α+1) ∫ Ω̄ε u(α+1) dx dy. Here we have used property (A10), which implies that ϕε(tu) → −∞ as t → +∞, and conclusion (ii) follows. � Now, let cε denote the mountain pass level associated with ϕε, that is cε = inf η∈Γ max t∈[0,1] ϕε(η(t)), where Γ = {η ∈ C([0, 1], Xε) : η(0) = 0 and ϕε(η(1)) < 0}. Lemma 2.2. If the conditions (A1), (A9), (A11)-(A14) hold, then, for each u ∈ Xε with u 6= 0, there exists a unique tε = tε(u) > 0 such that tεu ∈ Nε and ϕε(tεu) = maxt≥0 ϕε(tu). Moreover, cε = inf u∈Xε\{0} max t≥0 ϕε(tu) = inf Nε ϕε. (2.13) Proof. For a fixed w ∈ Xε \ {0}, let l(t) = ϕε(tw) for all t > 0. By the proof of Lemma 2.1, we observe that l(0) = 0, l(t) > 0 for t small enough and l(t) > 0 for t sufficiently large. Thus, there exists tε > 0 such that l(tε) = ϕε(tεw) = max t≥0 ϕε(tw) and l′(tε) = 0. Therefore, tεw ∈ Nε. Then we will prove the uniqueness of tε, and we have ‖w‖2ε = t(α−1) ε ∫ R2 g(εx, εy, tεw) (tεw)α w(α+1) dx dy. Hence, we define a function q : R+ 7→ [0,+∞] by q(t) := t(α−1) ∫ R2 g(εx, εy, tw) (tw)α w(α+1) dx dy. By calculation and condition (A14), we prove that the function q is increasing on (0,+∞). Therefore, tε is unique. The conclusion cε = inf w∈Xε\{0} max t≥0 ϕε(tw) is obvious and the proof is complete. � Lemma 2.3 ([1]). For any fixed ε > 0, the functional ϕε satisfies the (PS)c con- dition. Lemma 2.4. The functional ϕε has a nontrivial critical point uε ∈ Xε such that ϕε(uε) = cε and ϕ′ε(uε) = 0. 8 Z. XIE, J. CHEN EJDE-2023/48 Lemmas 2.1 and 2.3 permit us to apply the Mountain Pass Lemma due to Am- brosetti and Rabinowitz [25] to conclude that cε is a critical value for ϕε. Lemma 2.5. Suppose that (A1), (A4), (A5), (A7), (A9) hold. Then, for ε > 0 small, the functional ϕε restricted to Nε satisfies the (PS) condition. And the critical points of the functional ϕε on Nε are critical points of ϕε in Xε. Proof. Let {wn} ⊂ Nε be such that ϕε(wn) → c and ϕ′ε(wn)|Nε → 0. Then there exists {µn} ⊂ R satisfying ϕ′ε(wn) = µnΦ′ε(wn) + on(1), (2.14) where Φε : Xε → R is Φε(w) = 〈ϕ′ε(w), w〉 = ‖w‖2ε − ∫ R2 g(εx, εy, w)w dxdy. From 〈ϕ′ε(wn), wn〉 = 0, we have ‖wn‖2ε = ∫ R2 g(εx, εy, wn)wn dx dy. (2.15) By f ∈ C1(R,R) and the definition of g, we see that g is of C1. Then, by calcula- tions, we have 〈Φ′ε(wn), wn〉 = 2‖wn‖2ε − ∫ R2 [g′(εx, εy, wn)w2 n + g(εx, εy, wn)wn] dx dy = 2 ∫ R2 g(εx, εy, wn)wn dx dy − ∫ R2 [g′(εx, εy, wn)w2 n + g(εx, εy, wn)wn] dx dy = ∫ R2 [g(εx, εy, wn)wn − g′(εx, εy, wn)w2 n] dx dy. (2.16) Let R2 = Ω1 ∪ Ω2, where Ω1 = [Ωε ∪ {wn < t0}] ∪ [(R2 \ Ωε) ∩ {t0 ≤ wn ≤ t1}], Ω2 = [(R2 \ Ωε) ∩ {wn ≥ t1}]. On the one hand, for (x, y) ∈ Ω1, from (A14) we have g(εx, εy, wn)wn + g′(εx, εy, wn)w2 n ≥ 0, g(εx, εy, wn)wn − g′(εx, εy, wn)w2 n ≤ 0, One the other hand, for (x, y) ∈ Ω2, we have g(εx, εy, wn) = V0 k wn, and then g(εx, εy, wn)wn + g′(εx, εy, wn)w2 n = 2V0 k w2 n, ∀(x, y) ∈ Ω2. Since {wn} is bounded, up to a subsequence, we may suppose that 〈Φ′ε(wn), wn〉 → a ≤ 0. If a = 0, by the Sobolev continuous embedding Xε ↪→ Lp(R2) for p ∈ [2, 6] and choose some C2 < k V0 , we obtain |〈Φ′ε(wn), wn〉| ≥ 2‖wn‖2ε − 2V0 k ‖wn‖22 EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 9 ≥ (2− 2V0 k C2)‖wn‖2ε ≥ (2− 2V0 k C2) ∫ Ωε ( |wn|2 + |D−1 x (wn)y|2 + V (εx, εy)w2 n ) dx dy. Therefore, wn → 0 in Xε, wn → 0 in Lp(R2), wn → 0 in Lp(Ωε). Finally, by (A11)–(A14), we have ‖wn‖2ε = ∫ R2\Ωε g(εx, εy, wn)wn dx dy + on(1) ≤ V0 k ‖wn‖22 + on(1). (2.17) From the proof of Lemma 2.1, we can easy to check that there exists a number d > 0 such that ‖wn‖ε ≥ d > 0 for all wn ∈ Nε. (2.18) Thus, there is a contradiction between (2.16) and (2.17). We obtain a 6= 0 and µn = on(1). From (2.13), we conclude that ϕ′ε(wn) → 0, that is, {wn} is a (PS) sequence for ϕε. This completes the proof. � 3. 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 Mδ ⊂ Ω, where Ω is given in the condition (A9). We start by considering the limit problem associated with (2.10), namely, the problem( −uxx +D−2 x uyy + V0u− f(u) ) x = 0, in R2 (3.1) The solutions of (3.1) are precisely critical points of the functional defined by ϕ0(u) = 1 2 ∫ R2 [|ux|2 + |D−1 x uy|2 + V0u 2] dx dy − ∫ R2 F (u) dx dy (3.2) which is C1 with Gateaux derivative 〈ϕ′0(u), v〉 = ∫ R2 ( uxvx +D−1 x uyD −1 x vy + V0uv ) dx dy − ∫ R2 f(u)v dx dy, (3.3) for all u, v ∈ X0. Let N0 be the Nehari manifold associated with ϕ0 by N0 = {u ∈ X0 \ {0} : 〈ϕ′0(u), u〉 = 0}, (3.4) where X0 is defined as the Hilbert space Xε but endowed with the inner product (u, v)0 = ∫ R2 ( uxvx +D−1 x uyD −1 x vy + V0uv ) dx dy, (3.5) and the corresponding norm ‖u‖0 = (∫ R2 ( |ux|2 + |D−1 x uy|2 + V0u 2 ) dx dy )1/2 . (3.6) As in the previous section, the next lemma characterizes of the infimum of ϕ0 over N0. 10 Z. XIE, J. CHEN EJDE-2023/48 Lemma 3.1. If the conditions (A1), (A4), (A5), (A9) hold, then, for each u ∈ X0 with u 6= 0, there exists a unique t0 = t0(u) > 0 such that t0u ∈ N0 and ϕ0(t0u) = maxt≥0 ϕ0(tu). Moreover, we have cV0 = inf u∈X0\{0} max t≥0 ϕ0(tu) = inf N0 ϕ0. (3.7) where cV0 is the minimax level of Mountain Pass Theorem applied to ϕ0, namely cV0 = inf η∈Γ max t∈[0,1] ϕ0(η(t)), where Γ = {η ∈ C([0, 1], X0) : η(0) = 0 and ϕ0(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 (PS)cV0 sequence is nontrivial. Lemma 3.2. If {wn} is bounded in Xε and there exist R > 0 and a sequence (xn, yn) ∈ R2 such that lim sup n→∞ ∫ BR((xn,yn)) |wn(x, y)|2 dx dy = 0. Then wn → 0 strongly in Lq(R2), for every 2 < q < 6. Proof. First of all, we fix q ∈ (2, 6). Given R > 0 and z ∈ R2, by standard interpolation inequality and Sobolev embedding theorem, we obtain ‖wn‖Lq(BR(z)) ≤ ‖wn‖1−λL2(BR(z))‖wn‖ λ L6(BR(z)) ≤ C3‖wn‖1−λL2(BR(z)) (∫ BR(z) [|(wn)x| 2 + |D−1 x (wn)y| 2 + V (εx, εy)wn 2] dx dy )λ/2 , where 1−λ 2 + λ 6 = 1 q . Now, covering R2 with balls of radius R, in such a way that each point of R2 is contained in at most 3 balls, we find that∫ R2 |wn|q ≤ 3C3 sup z∈R2 [ ∫ BR(z) wn 2 ] (1−λ)q 2 ‖wn‖λq/2ε . Under the assumption of the lemma, we have wn → 0 in Lq(R2). � Lemma 3.3 ([10]). Let {wn} ⊂ X0 be a (PS)cV0 sequence for ϕ0 and such that wn ⇀ 0 in X0. Then, only one of the following alternatives holds. (i) wn → 0 in X0, or (ii) there exist a sequence {(xn, yn)} ⊂ R2 and constants R, β > 0 such that lim inf n→∞ ∫ BR((xn,yn)) |wn|2 dx dy ≥ β > 0. Lemma 3.4. Let cV0 > 0 and {wn} be a (PS)cV0 sequence for ϕ0, then {wn} is bounded. Proof. Assume that {wn} ⊂ X0 is a (PS)cV0 sequence for ϕ0. Then ϕ0(wn)→ cV0 and ϕ′0(wn)→ 0. EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 11 Therefore, there exists C4 > 0 large enough such that C4(1 + ‖wn‖0) ≥ ϕ0(wn)− 1 α+ 1 〈ϕ0 ′(wn), wn〉, ∀n ∈ N (3.8) On the other hand, by conditions (A10) and (2.9), we have ϕ0(wn)− 1 α+ 1 ϕ0 ′(wn)wn = ( 1 2 − 1 α+ 1 )‖wn‖20 + ∫ R2 [ 1 α+ 1 f(wn)wn − F (wn)] dx dy ≥ ( 1 2 − 1 α+ 1 )‖wn‖20. The above inequality proves that {wn} is bounded in X0. � Theorem 3.5. Suppose that (A1), (A4), (A5), (A7), (A9), (A10) are satisfied. Then problem (3.1) has a positive ground-state solution. Proof. Let {wn} ⊂ X0 be a (PS)cV0 sequence for ϕ0. By Lemma 3.4, we know that {wn} is bounded in X0. Then, up to a subsequence, wn ⇀ w weakly in X0, wn → w strongly in Lploc(R2), p ∈ [2, 6) and wn → w a.e. in R2. As in the proof of Lemma 2.1, we can easily prove that ϕ0 satisfies the Mountain Pass Geometry. By the Mountain Pass Lemma (see [25]), there exists a Palais-Smale sequence {wn} for ϕ0 at the mountain pass level cV0 . Moreover, wn ⇀ w in X0 and w is a critical point of ϕ0. From Lemma 3.2, we know that w 6= 0 and w ∈ N0. Next we prove that wn → w strongly in X0. From the semi-lower continuity of norm, we have lim n→∞ ‖wn‖0 ≥ ‖w‖0. (3.9) Observe that we must have the above equality hold. Otherwise, by Fatou’s Lemma we obtain cV0 ≤ ϕ0(w)− 1 α+ 1 〈ϕ′0(w), w〉 = (1 2 − 1 α+ 1 ) ‖w‖20 + 1 α+ 1 ∫ R2 [f(w)w − (α+ 1)F (w)] dx dy < lim inf n→∞ { ( 1 2 − 1 α+ 1 )‖wn‖20 + 1 α+ 1 ∫ R2 [f(wn)wn − (α+ 1)F (wn)] dx dy } = lim inf n→∞ ( ϕ0(wn)− 1 α+ 1 〈ϕ′0(wn), wn〉 ) = cV0 . which is a contradiction. Thus, we conclude that up to a subsequence, lim n→∞ ‖wn‖0 = ‖w‖0. (3.10) So, the Brezis-Lieb Lemma [6] implies that wn → w in X0. Last, we prove that the solution w is nonnegative. From (A7) and using −w− as a testing function, we have ϕ′0(w)(−w−) = ∫ R2 ( |w−x |2 + |D−1 x w−1 y |2 + V0|w−|2 ) dx dy = ‖w−‖20 = 0. (3.11) where w− = max{−w, 0}. This implies that w ≥ 0 in R2 is a nonnegative weak solution of (3.1) and the proof is complete. � 12 Z. XIE, J. CHEN EJDE-2023/48 Then we consider δ > 0 such that Mδ ⊂ Ω and choose η ∈ C∞0 (R+, [0, 1]) satisfying η(s) = { 1, if 0 ≤ s ≤ δ/2, 0, if s > δ. For each z ∈M = {x ∈ Ω : V (x) = V0}, let ν ∈ R2, we define ψε,z(ν) = η(|(εν − z)|)w( εν − z ε ) and tε > 0 satisfying max t≥0 ϕε(tψε,z) = ϕε(tεψε,z), where w is a solution of (3.1) such that ϕ0(w) = cV0 . We define φε : M → Nε by φε(z) = tεψε,z. Lemma 3.6. The function φε satisfies lim ε→0 ϕε(φε(z)) = cV0 uniformly in z ∈M. Proof. Arguing by contradiction, we suppose that there exist δ > 0, {zn} ⊂ M , and εn → 0, such that |ϕεn(φεn(zn))− cV0 | ≥ δ. (3.12) Notice that for each n ∈ N and for all ν ∈ Bδ/εn( znεn ), we have εnν ∈ Bδ(zn). By using the change of variables ν̄ := (εnν − zn)/εn, one easily has εnν̄ + zn ∈ Bδ(zn) ⊂Mδ ⊂ Ω. (3.13) and ϕεn(φεn(zn)) = ϕεn(tεnψεn,zn) = t2εn 2 ‖ψεn,zn‖2ε − ∫ R2 G(εnx, εny, tεnψεn,zn) dx dy = t2εn 2 ‖η(|εnν̄|)w(ν̄)‖2ε − ∫ R2 F (tεnη(|εnν̄|)w(ν̄)) dx dy. (3.14) Since φεn(zn) ∈ Nεn and g = f in Ω, we have t2εn‖η(|εnν̄|)w(ν̄)‖2εn = ∫ R2 g (εnν̄ + zn, tεnη(|εnν̄|)w(ν̄)) tεnη(|εn|)w(ν̄) dx dy = ∫ R2 f (tεnη(|εnν̄|)w(ν̄)) tεnη(|εn|)w(ν̄) dx dy. (3.15) By (A5), if tεn →∞, we obtain ‖ψεn,zn‖2εn = ∫ R2 g(εnx, εny, tεnψεn,zn) tεnψεn,zn ψ2 εn,zn dx dy > ∫ Bδ/2(0) C0(tεnψεn,zn)q−2ψ2 εn,zn dx dy = ∫ Bδ/2(0) C0t q−2 εn (η(|εnν̄|)w(ν̄))q dx dy ≥ C0t q−2 εn cq|Bδ/2(0)| → ∞. (3.16) EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 13 as tεn →∞. Where c = infu∈Bδ/2(0) w(u), and q ∈ (2, 6) is given in (A5). But the left side of the above inequality is bounded as tεn →∞. This yields a contradiction. Hence, tεn → t0 with t0 ≥ 0. Form (3.15) and (A4), we see that t0 > 0. Now we prove a claim. Claim. t0 = 1. By using Lebesgue’s theorem, we can verify that lim n→∞ ‖ψεn,zn‖2εn = ‖w‖20, lim n→∞ ∫ R2 ψεn,zn = ∫ R2 w dxdy, lim n→∞ ∫ R2 f(ψεn,zn)ψεn,zn = ∫ R2 f(w)w dxdy. Therefore, passing to limit in equality (3.15), we have ‖w(ν̄)‖2ε = lim n→∞ ∫ R2 g(εnν̄, tεnψεn,zn) tεnψεn,zn ψ2 εn,zn dx dy = lim n→∞ ∫ R2 g(εnν̄, tεnη(|εnν̄|)w)(η(|εnν̄|)w(ν̄))2 tεnη(|εnν̄|)w(ν̄) dx dy = ∫ R2 f(t0w) t0w w2 dx dy. (3.17) Since w is a positive ground state of (1.2), we have that 〈I ′ε(w), w〉 = 0, namely ‖w‖2ε = ∫ R2 f(w)w dxdy. (3.18) Combining (3.17) and (3.18), we have tα−1 0 ∫ R2 f(t0w) (t0w)α wα+1 dx dy = ∫ R2 f(w) wα wα+1 dx dy. (3.19) Arguing as in the proof of uniqueness in Lemma 2.2, we see that t0 = 1 and we obtain the claim. Letting n→∞ in (3.14), we obtain limεn→0 ϕεn(φεn(zn)) = ϕ0(w) = cV0 , which contradicts to (3.12). This completes the proof. � 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. Proposition 3.7 ([1]). Let εn → 0 and {wn} ⊂ Nεn be such that ϕεn(wn)→ cV0 . Then there exists a sequence {z̃n} ⊂ R2 such that vn(x) = wn(x + z̃n) has a convergent subsequence in X0. Moreover, up to a subsequence, zn := εnz̃n → z0 ∈ M . We now consider the following subset of the Nehari mainfold N cV0+h(ε) ε = {w ∈ Nε : ϕε(w) ≤ cV0 + h(ε)}, where h(ε) = |ϕε(φε(z)) − cV0 | is such that h(ε) → 0+ as ε → 0+. Thus, there is a number ε̂ > 0 such that for all ε ∈ (0, ε̂), we have φε(z) ∈ N cV0+h(ε) ε and N cV0+h(ε) ε 6= ∅. 14 Z. XIE, J. CHEN EJDE-2023/48 Let ρ = ρ(δ) > 0 be such that Mδ ⊂ Bρ(0). We define χ : R2 → R2 as χ(x, y) = (x, y) for |(x, y)| ≤ ρ and χ(x, y) = ρ(x, y)/|(x, y)| for |(x, y)| ≥ ρ. Let us consider the barycenter map βε : Nε → R2 be given by βε(w) = ∫ R2 χ(εx, εy)w2(x, y) dx dy∫ R2 w2(x, y) dx dy . Since M ⊂ Bρ(0), by the definition of χ and the Lebesgue’s theorem, we have lim ε→0 βε(φε(z)) = z uniformly in z ∈M. (3.20) Lemma 3.8. For any δ > 0, it holds that lim ε→0 sup w∈N cV0 +h(ε) ε dist(βε(w),Mδ) = 0. The proof of the above lemma is similar to the proof of [13, Lemma 3.10]. We omit its proof. Theorem 3.9. Assume that conditions (A1), (A9), (A4), (A5), (A7), (A10) hold. Then for each δ > 0, there exists ε̂δ > 0 such that, for any ε ∈ (0, ε̂δ), problem (2.10) have at least catMδ (M) positive solutions. Proof. From Lemmas 3.6 and 3.8, and (3.20), we see that the continuous mappings below is well defined for ε ∈ (0, ε̂), M φε→ N cV0+h(ε) ε βε→Mδ. From (3.20), we can choose a function θ(ε, z) with |θ(ε, z)| < δ 2 uniformly in z ∈M , such that βε(φε(z)) = z + θ(ε, z) for all ε ∈ (0, ε̂), z ∈M. We define H(t, z) = z + (1 − t)θ(ε, z). Then H : [0, 1] ×M → Mδ is continuous. Obviously, H(0, z) = βε(φε(z)), H(1, z) = z for all z ∈ M . That is, H(t, z) is a homotopy between βε ◦ φε and the inclusion map Id : M →Mδ. From the proof of [20, Theorem 1.1], it is easy to check that there exists a subset A ⊂ Nε and we select suitably c̄ > cV0 + h(ε) such that (i) A is not contractible in N cV0+h(ε) ε , (ii) A is contractible in N c̄ ε . The above properties imply that there exists a critical level between cV0 +h(ε) and c̄. Therefore, there exists a number ε̂δ > 0 small, then for ε ∈ (0, ε̂δ), it follows from Lemma 2.5 that the functional ϕε restricted to Nε satisfies the (PS) condition in the interval (cV0 , cV0 +h(ε)). Notice that the well known properties of the category in [23] guarantee cat(N cV0+h(ε) ε ) ≥ catMδ (M), and the Ljusternik-Schnirelmann theory ensures the existence of at least cat(N cV0+h(ε) ε ) critical points of ϕε in N cV0+h(ε) ε . Therefore, we conclude that ϕε admits at least catMδ (M) critical points inN cV0+h(ε) ε . Moreover, from the proof of Theorem 3.5, we infer that these solutions of problem (2.10) are positive by a similar argument. � EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 15 4. 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 uε ≤ a,∀x ∈ R2 \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. Lemma 4.1. Let εn → 0+ and wn ∈ N cV0+h(εn) εn be a solution of problem (2.10). Then, up to a subsequence, there exists a sequence (xn, yn) ⊂ R2 such that ψn(x, y) := wn(xn + x, yn + y) satisfies that wn(x, y) = ψn(x, y) ∈ L∞(R2), and there exists C5 > 0 such that ‖wn‖L∞(R2) ≤ C5, for all n ∈ N, (4.1) where (xn, yn) = z̃n 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 Mδ ⊂ Ω. Similar to the proof of [13, Theorem 1.1], there exists ε̃δ > 0 such that for any ε ∈ (0, ε̃δ) and b > 0, then for any solution wε ∈ N cV0+h(ε) ε of problem (2.10), it holds ‖wε‖L∞(R2\Ωε) < b. (4.2) Let ε̂δ be given in Theorem 3.9 and ε := min{ε̂δ, ε̃δ}. From Theorem 3.9, we can know that there exists catMδ (M) nontrivial solutions of problem (2.10) in N cV0+h(ε) ε . If w ∈ Xε is one of these solutions, then w ∈ N cV0+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 catMδ (M) nontrivial solutions. The proof of Theorem 1.1 is complete. � Acknowledgments. The authors would like to thank the referees for their useful suggestions. 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Ding; Stationary solutions for a generalized Kadomtsev-Petviashvili equa- tion in bounded domain, Electronic Journal of Qualitative Theory of Differential Equations. 2012 (2012), No. 68, 1-18. [30] W. M. Zou; Solitary waves of the generalized Kadomtsev-Petviashvili equations, Appl. Math. Lett. 15 (2002), 35-39. [31] Y. Zhu, C. Chen, J. Chen, et al; Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation, Open Mathematics. 19 (2021), 297-305. Zheng Xie School of Mathematics and Computer science, Hunan University of Science and Tech- nology, Xiangtan, 411201 Hunan, China Email address: xz@mail.hnust.edu.cn EJDE-2023/48 SOLUTIONS TO KADOMTSEV-PETVIASHVILI EQUATIONS 17 Jing Chen (corresponding author) School of Mathematics and Computer science, Hunan University of Science and Tech- nology, Xiangtan, 411201 Hunan, China Email address: cjhnust@hnust.edu.cn 1. Introduction 2. Variational framework 3. Multiplicity of solutions for the modified problem 3.1. Autonomous problem 3.2. Multiplicity of solutions for (2.10) 4. Proof of Theorem 1.1 Acknowledgments References