Multiplicity of Solutions for a Class of Kirchhoff – Poisson Type Problem

the


Introduction and Main Results
This paper is concerned with the existence and multiplicity of solutions for the nonlinear Kirchhoff-Poisson type problem: where Ω is a smooth bounded domain in R N (N ¼ 1; 2 or 3), a; b>0 and f : Ω × RÀ!R is a continuous function, satisfying some suitable conditions we will formulate later.When a ¼ 1 and b ¼ 0; Equation (1) reduces to the boundary value problem: Knowledge of the solutions of System ( 2) is related to the study of stationary solutions ψðx; tÞ: ¼ e −it uðxÞ: for the nonlinear parabolic Schrödinger-Poisson system: The first equation in Problem (3) is called Schrödinger equation, and modeling quantum nonrelativistic particles interacting with the eletromagnetic field are caused by the motion.A typical and important class of Schrödinger equations is reflected in the potential ϕðxÞ: , which is depended on the charge of wave function itself, that is to say, as the Poisson equation in Equation (3) holds.For more applications of the physical relevance of the Schrödinger-Poisson system, we refer to [1][2][3][4] and references therein.System (2) has been extensively studied after the seminal work of Benci and Fortunato [4].There are many results about the existence and nonexistence of solutions, multiplicity of solutions, least energy solutions, radial and nonradial solutions, semiclassical limit, and concentrations of solution which are covered in the literature (see for instance [1][2][3][5][6][7][8][9][10][11], and the references therein).
When the potential ϕ in Equation ( 1) vanishes, we get the following problem: In the case b ≠ 0; Problem (4) is nonlocal due to the emergence of b R Ω jruj 2 dxΔu and is involved in the following stationary analog of equation: which was proposed by Kirchhoff as a generalization of the classical D'Alembert's wave equation for the free vibration of elastic strings: where L is the length of the string, h denotes the area of crosssection, E represents the Young modulus of the material, and P 0 is recorded as the initial tension.The Kirchhoff's model deals with the change of string length caused by transverse vibrations, we refer to [12][13][14] for early work.It is wellknown that only after Lions [15] put forward an abstract analysis framework, Equation ( 6) attracted great attention, see [16][17][18] for example.Recently, many nonlinear analytical methods and techniques are employed to investigate the existence of sign-changing solutions to Problem (4) or similar Kirchhoff-type equations, and consequently, some interesting results were obtained.Let me cite a few examples, Alves et al. [19] and Ma and Rivera [20] obtained the existence of positive solution to this kind of problem by means of variational method.Perera and Zhang [21] utilized the method of Yang index and critical group to obtain the nontrivial solution of Equation (4).Also in [22], they revisited Equation ( 4) by means of the invariant set of descending flow and the existence of signed solutions and sign-changing solutions is considered.Analogous results were established in [23] by Mao and Zhang [23].The authors in [24][25][26][27] studied Problem (4) or more general Kirchhoff-type equations, respectively, by using constraint variational methods and quantitative deformation lemma.Later, under some more weaken assumptions on f , with the aid of some new analytical skills and non-Nehari manifold method, Tang and Cheng [28] extended some results obtained in [26].It is well-known that similar nonlocal problems can be used to model some physical and biological systems, where u describes a process depending on its own average value, such as population density, see [29][30][31] for example.Inspired ed by the papers mentioned above, the main goal of this paper is to show the existence and multiplicity of nontrivial solutions to Problem (1).The main tool is based on the classical fountain theorem in [32] and a variant version of the fountain theorem from Zou's [33] study.To the best of our knowledge, among the existing literatures, there is no such kind of result concerned with infinitely many solutions for Problem (1).
In this paper, we introduce the space W ¼ H 1 0 ðΩÞ: endowed with the norm kujj 2 : ¼ R Ω jruj 2 dx.Throughout this paper, we denote by j⋅j q , the usual L q -norm with q ≥ 1.Since Ω is a bounded domain, it is well-known that W↪L q ðΩÞ: continuously for q 2 ½1; 2 * : ; and compactly for q 2 ½1; 2 * Þ: : Moreover, there exists c q >0 such that: We set by 0<λ 1 <λ 2 <…, the distinct eigenvalues of − Δ in L 2 ðΩÞ: with zero Dirichlet boundary conditions and denote by e 1 ; e 2 ; e 3 ; … the eigenfunctions corresponding to eigenvalues, respectively.
It is well-known that, by the Lax-Milgram theorem, for every u 2 H 1 0 ðΩÞ: ; there exists a unique element ϕ u 2 H 1 0 ðΩÞ: such that: It is clear that the energy functional associated with Problem (1) can be expressed as follows: where Fðx; tÞ: ¼ R t 0 f ðx; sÞ: ds.It is not difficult to check that Γ is of class C 1 : and the critical points of Γ correspond to the weak solutions of Equation ( 1).Now, we state our main results as follows.
Theorem 1.We assume the nonlinearity f satisfies the following conditions: ( : and some constant C >0 such that: ðF 2 Þ: There exist constants θ >4 and R>0 such that: Abstract and Applied Analysis Then Problem (1) admits a sequence of solutions ω n f g: such that Γðω n Þ: À!1 as nÀ!1.
Then, Problem (1) admits infinitely many solutions ω n f g: Remark 1.In order to prove Theorem 1, we shall use the Ambrosetti-Rabinowitz type 4-superlinear condition ðF 2 Þ: to obtain the boundedness of (PS) sequences of the functional Γ.But, there are many functions which are 4-superlinear growth; however, it does not satisfy ðF 2 Þ: for any θ >4; hence, when ðF 2 Þ: is not verified, it becomes more complicated to deal with.In Theorem 2, we employ Theorem 4 without (PS)-type assumption, to establish arbitrarily many solutions of Equation ( 1) under some weaker conditions than ðF 2 Þ: .We present a concrete example at the end of the proofs to explain the main results.

Proofs of Theorems 1 and 2
We first recall the following preliminary results, which are a collection of results from D'Aprile and Mugnai's [34] and Ruiz and Siciliano's [7] studies.
For j ≥ 2, we set the following equations: We also need the following variant version of the fountain theorem.
Proof of Theorem 1.It is obvious that ΓðuÞ: 2 C 1 ðW; RÞ: .We have from ðF 2 Þ: by integrating that: Abstract and Applied Analysis c t Take ξ 2 ðθ −1 ; 4 −1 Þ: and ω n f g: ⊂ W, a ðPSÞ c -sequence of Γ.Thus, for n large enough, by Lemma 1-(i), we infer that: where C; c>0 are constants, and ω n ¼ y n þ z n , y n 2 Y n , and z n 2 Z n .From the fact that dim Y n is finite, and all norms in Y are equivalent, we see that ω n f g: is bounded in W. Therefore, there exists a subsequence of ω n f g: , still denoted by itself, such that ω n ⇀ω in H 1 0 ðΩÞ: .Thus, by using the Rellich theorem, we have ω n À!ω in L p ðΩÞ: ; also we can infer that f ðx; ω n Þ: À!f ðx; ωÞ: in L p ðΩÞ: with p ¼ q q−1 as a consequence of Theorem A.2 [32].Next, we prove that ω n f g: admits a convergent subsequence.Notice that: from which, we get the following equation: Abstract and Applied Analysis It is obvious that the first term of the right-hand side of Equation ( 25) converge to zero as nÀ!1, by virtue of ω n ⇀ω in E and the boundedness of ω n f g: in W. For the second term of the right-hand side of Equation ( 25), we have by using Hölder inequality that: where jϕ ω n − ϕ ω j 3 À!0 due to Lemma 1. Again by Hölder inequality, we get the following equation: when nÀ!1.Therefore, the right-hand side of Equation ( 25) tends to zero, and so we infer to kω n − ωjj : À!0, as nÀ!1.This implies condition ðD 3 Þ: is satisfied.Using Equation ( 22), we obtain the following equation: As all norms are equivalent on the finite dimensional space Y k , ðD 1 Þ: is satisfied for every sufficiently large ρ k >0.
We next verify condition ðD 2 Þ: .By ðF 1 Þ: , we have the following equation: Define such that on Z j , we have the following equation: As in [32], we can infer to β j À!0, jÀ!1, for any ω 2 Z j with kωjj: ¼ γ j , and so, we have the following equation: as jÀ!1.Thus, condition ðD 2 Þ: is satisfied.Now, we have checked that all the conditions of Theorem 3 hold; hence, Problem (1) admits a sequence of solutions ω j È É : such that Γðω j Þ: À!1 and jÀ!1.
On the other hand, by conditions ðF 1 Þ: , ðH 2 Þ: , we have, for any ε>0, there exists D ε >0, such that: Let β j be defined as Equation (30).Then, for each ω 2 Z j and ε>0 small enough, we get the following equation: Denote by γ j ¼ ð a Þ 1 2−q , then for ω 2 Z j with kωjj: ¼ γ j , one has the following equation: which implies that b j ðλÞ: ¼ inf ω2Z j ; kωjj¼γ j Γ λ ðωÞ: ≥ b * j À!1 as jÀ!1.Therefore, by Theorem 4, for a.e.λ 2 ½1; 2: , there exists a sequence ω such that: and as nÀ!1.Furthermore, using the fact that c j ðλÞ: ≤ sup ω2B j Γ λ ðωÞ: : ¼ c * j , and H 1 0 ðΩÞ: is imbedded compactly into L r ðΩÞ: for 2 ≤ r <2 * , by a standard argument, we infer that ω has a convergent subsequence.Consequently, there exist z j ðλÞ: such that Γ 0 λ ðz j ðλÞÞ: ¼ 0 and Γ λ ðz j ðλÞÞ: 2 ½b * j ; c * j : .As a result, we can find λ n À!1 such that z n f g: being exactly what kind of want to happen.Claim 2. z n f g 1 n¼1 must be bounded in W. Suppose by contradiction that, kz n jj: À!1 as nÀ!1.Denote by u n : ¼ z n kz n jj .Then up to a subsequence, we have the following equation: u n À!u a:e: x 2 Ω: There are two possible cases: In Case (i), it follows from Γ 0 λ n ðz n Þ: ¼ 0 and Lemma 1 that: On the other hand, by Fatou's lemma and condition ðH 1 Þ: , ðH 3 Þ: , we infer to the following equation: which yields a contradiction.In Case (ii), we may define the following functional as in [35]: Abstract and Applied Analysis Set u * n : ¼ ð4ℓÞ 1 2 u n with ℓ>0, then we obtain when n is large enough, that: which implies that lim nÀ!1 Γ λ n ðt n z n Þ: ¼ 1, since ℓ>0 can be large arbitrarily.Here, we have used the fact that u n ⇀0 in W and u n À!0 in L p ðΩÞ: ; p 2 ½1; 6Þ: ; and so: Note that t n 2 ð0; 1: and hΓ 0 λ n ðt n z n Þ; t n z n i: ¼ 0, it follows from: and Γ λ n ð0Þ: ¼ 0, that jt n z n j: must tend to 1 when nÀ!1.Therefore, by ðH 3 Þ: and λ n 2 ½1; 2: , we have the following equation: On the other hand, we use the fact that hðtÞ: ¼ t 4 f ðx; sÞ : s − 4Fðx; tsÞ: is increasing in t 2 ð0; 1: , which implies that, f ðx; sÞ: s − 4Fðx; sÞ: is increasing in s>0 by virtue of the following equation: and f ðx; sÞ s 3 is increasing for s 2 ð0; 1Þ: .By virtue of the oddness of f , we have the following equation: which leads to a contradiction in view of: So far, we have proved that the solution z j satisfies Γðz j Þ : 2 ½b * j ; c * j : .Since b * j À!1 as jÀ!1, we see that there exist a sequence of solutions z j f g 1 j¼1 of Problem (1) such that Γ 1 ðz j Þ : À!1, jÀ!1.
Finally, we present an example to explain that there is a nonlinear f which satisfies all the conditions of Theorem 2, but does not satisfy the conditions of Theorem 1, especially condition ðF 2 Þ: .Example 1.Let f ðx; uÞ: ¼ u 3 lnð1 þ jujÞ: : Integrating by parts, a simple computation yields that for u ≥ 0; we have the following equation: as juj: À!1; which means that condition ðH 3 Þ: is satisfy.Moreover, it is easy to see that f satisfies conditions ðH 1 Þ: ; ðH 2 Þ: : Abstract and Applied Analysis However, f does not satisfy condition ðF 2 Þ: .Indeed, suppose that there exists some μ>4 fulfilling θFðx; uÞ: ≤ f ðx; uÞ: u for juj: large.Consequently, we have the following equation: for juj: large, which contradicts to the fact θ >4: Data Availability