Multiplicity of Solutions for a Modified Schrödinger-Kirchhoff-Type Equation in R N

proposed by Kirchhoff in [1] as an existence of the classical D’Alembert’s wave equation for free vibrations of elastic string. Kirchhoff ’s model takes into account the changes for free vibrations of elastic strings. Recently, there have been many papers concerned with the Kirchhoff-type problems by variational methods; see [2–8] and the references therein. Many studies of them are concentrated on a bounded smooth domain Ω of R (N = 1, 2, 3); it is well known that the embedding H(R) 󳨅→ L(R) (2 ≤ t < 2) is not compact. Hence, if we look for solution by variational methods, it is very difficult to prove (PS) condition. Moreover, in order to check the (PS) condition or some of its variants, one has to impose certain conditions. When b = 0 and a = 1, (1) is reduced to the following modified nonlinear Schrödinger equation:


Introduction and Main Result
In this paper, we study the following modified Schrödinger-Kirchhoff-type equations of the form where  > 0,  ≥ 0,  ≥ 3,  ∈ (R  × R, R), and  ∈ (R  , R).
When Δ 2 = 0, (1) is reduced to the following Kirchhoff-type problem: (2) If () = 0, problem (2) is related to the stationary analogue of the Kirchhoff equation proposed by Kirchhoff in [1] as an existence of the classical D' Alembert's wave equation for free vibrations of elastic string.Kirchhoff's model takes into account the changes for free vibrations of elastic strings.Recently, there have been many papers concerned with the Kirchhoff-type problems by variational methods; see [2][3][4][5][6][7][8] and the references therein.Many studies of them are concentrated on a bounded smooth domain Ω of R  ( = 1, 2, 3); it is well known that the embedding  1 (R  ) →   (R  ) (2 ≤  < 2 * ) is not compact.Hence, if we look for solution by variational methods, it is very difficult to prove () condition.Moreover, in order to check the () condition or some of its variants, one has to impose certain conditions.When  = 0 and  = 1, (1) is reduced to the following modified nonlinear Schrödinger equation: Solutions of (4) are standing waves of the following quasilinear Schrödinger equations: + Δ −  ()  + Δ ( (          2 ))   (          2 )  +  (, ) = 0,  ∈ R N , where  is a real constant and  and  are real functions.Equations ( 5) are derived as model of several physical phenomena, such as [9,10].Many achievements had been obtained on the existence of ground states, infinitely many solutions, and soliton solutions for (4), by a dual approach, Nehari method, and the minimax methods in critical point theory, applying the perturbation approach and the Lusternik-Schnirelmann category theory; see [11][12][13][14][15][16][17][18].
In this paper, we transform (1) to another equation with a continuous energy functional in some Banach space.We obtain the existence of multiple solution for problem (1) via using nonsmooth critical point theory and using some new techniques.Throughout this paper, the main ideas used here come from Colin and Jeanjean [12] and Liu et al. [14].
We need the following several notations.Let with the inner product and the norm Let and the norm Let the following assumption () hold: Moreover, we need the following assumptions: ( 2 ) There exist constants  > 0 and 4 <  < 22 * such that for all (, ) ∈ R  × R, where 2 * = 2/( − 2) is the Sobolev critical exponent.
( Throughout the paper, → and ⇀ denote the strong and weak convergence, respectively., ,   , and   express distinct constants.For 1 ≤  < ∞, the usual Lebesgue space is endowed with the norm The paper is organized as follows.In Section 2, we reformulate our problem into a new one which has an associated functional well defined in a suitable space and present some preliminary results.In Section 3, we introduce some notions and results of nonsmooth critical point theory and we show that the functional  satisfies ()  condition.In Section 4, we complete the proof of Theorem 1.

The Dual Variational Framework and Preliminary Results
The energy functional corresponding to problem (1) is defined as follows: It should be pointed out that the main difficulty in treating this class of quasilinear equations in R  is the lack of compactness and the second-order nonhomogeneous term Δ 2 which prevents us from working directly in a classical function space. is not defined in  1  (R  ); thus we may not apply directly the variational method to study (1).To move these obstacles, we make the change of variable (see [12,14]); that is, we consider V =  −1 (), where  is defined by In order to prove our main result, we need some further properties of the function .
After this change of variables, the functional is well defined on which is a Banach space endowed with the norm A standard argument which is similar to that in [12] shows that if V ∈  is a critical point of the functional , then  = (V) ∈  and  is a weak solution of (1).
We have the following result with respect to the space .Its proof can be found in [13,14].
(2) For every V ∈  and  ∈  ∩  ∞ (R  ), the derivation of  in the direction  at V exists and will be denoted by (3) The map ⟨V, ⟩ → ⟨  (V), ⟩ satisfies the following: Remark 5. Consider the following: Proof.Let {V  } ⊂  be a sequence such that V  → V in .By Proposition 3, we have lim  Consequently, (V  ) → (V) and  is continuous in .

Nonsmooth Critical Framework
Let us begin by recalling some notions and results of nonsmooth critical point theory (see [21,22]).
In the following,  will denote a metric space endowed with the metric .We can state a generalized version of the symmetric Mountain Pass Theorem for the case of continuous functionals.
(ii) For every finite-dimensional subspace  ⊂ , there exists  > 0 such that Then there exists a sequence {  } of critical values of  with   → ∞ as  → ∞.
Now consider the functional  given in the previous section.
Proof.The proof is completed by Lemmas 10 and 11.
Proof.Let {V  } ⊂  be a sequence with (V  ) →  and ||(V  ) → 0.Then, it follows from Lemma 10 that Moreover, Note that Letting we deduce that ‖  ‖ ≤ ‖V  ‖ for some positive constant .
From ( 49) and (50), we have where  1 > 0. Their estimates imply that V  is bounded in .