Positive Solutions for a Class of Quasilinear Schrödinger Equations with Nonlocal Term

This paper is considered the existence of positive solutions for a class of generalized quasilinear Schrödinger equations with nonlocal term in N  which have appeared from plasma physics, as well as high-power ultrashort laser in matter. We use a charge of variables and obtain the existence of solutions via minimization


Introduction
In this paper, we consider investigating the existence of solutions for the follow-  appeared at least as early as in 1954, in a work by Pekar describing the quantum mechanics of a polaron at rest [1]. In 1976, Choquard used ( 1.3) to describe an electron trapped in its own hole and in a certain approximation to Hartree-Fock theory of one component plasma [2]. In 1996, Penrose proposed ( 1.3) as a model of self-gravitating matter, in a program in which quantum state reduction is understood as a gravitational phenomenon [3]. In this context, equation of type ( 1.3) is usually called the nonlinear Schrödinger-Newtonequation. The first investigations for the existence and symmetry of the solutions to (1.3) go back to the works of Lieb [2] and Lions [4]. In [2], by using symmetric decreasing rearrangement inequalities, Lieb proved that the ground state solution of Equation ( 1.3) is radial and unique up to translations. Lions [4] showed the existence of a sequence of radially symmetric solutions. Ma and Zhao [5] considered the generalized Choquard equation 1.4) and proved that every positive solution of it is radially symmetric and monotone decreasing about some fixed point, under the assumption that a certain set of real numbers, defined in terms of N, and q, is nonempty. Under the same assumption, Cingolani, Clapp, and Secchi [6] gave some existence and multiplicity results in the electromagnetic case and established the regularity and some decay asymptotically at infinity of the ground states. In [7], Moroz and Van Schaftingen eliminated this restriction and showed the regularity, positivity and radial symmetry of the ground states for the optimal range of parameters and derived decay asymptotically at infinity for them as well. Moreover, they [8] also obtained a similar conclusion under the assumption of Berestycki-Lions type nonlinearity. We point out that the existence, multiplicity, and concentration of such like equation have been established by many authors. We refer the readers to [9] [10] for the existence of sign-changing solutions, [11] [12] for the existence and concentration behavior of the semiclassical solutions and [13] for the critical nonlocal part with respect to the Hardy-Littlewood-Sobolev inequality. For more details associated with the Choquard equation, please refer to [14] [15] [16] and the references in. Li, Teng, Zhang, Nie [17] investigate the existence of solutions for the following generalized quasilinear Schrödinger equation with nonlocal term and prove the existence of solution.
In this paper, our main ideas come from [18] and the assumption of g from [19]. Our purpose is to search for the existence of nontrivial solutions of ( To overcome this difficulty, we make a change of variable constructed by Shen and Wang in [20]: We say that u is a weak solution of (1.1), if , by [20], we know that the above formula is equivalent to  . Therefore, in order to find the solution of (1.1), it suffices to study the solution of following equation: J is defined on the space is a Banach space. In the following, we always assume ( ) Let us consider the following assumptions of potential function ( ) Next, we will introduce the properties of some functions.
are strictly increasing and odd; Next, we set forth some preliminary results.

Preliminary Results
To begin with, we prove some functions are continuous, more detailed see [21].
Proof: By sobolev imbedding inequality, Lemma1.1 and definition of g, we Proof: By the definition of g, we have , from Lemma 3.4 [22], we get the result.  Next, we introduce some minimization with corresponding energy functional and define Therefore, we have following fact.
, from the definition of g, we get

( )
F v is well defined and continuous for 2 is a continuous function, and For (2) Using the definition of g and Lemma 1.1, we know ( ) For the third term, we have Similarly to above, by the dominated convergence theorem Finally, the continuity with strong-weak topology is easy to check, as 2.4 does not show that ( ) F v is C ∞ , so we cannot use the Lagrange multiplier theorem. But we can get our conclusion we want exactly by a similar argument for the Lagrange multiplier theorem. Next, we state our main conclusion. The idea of our proof is based on the work in [18] [22] [23]. is compact. In the process of the proof of theorem 3.1, it is important for us to construct auxiliary function, then by implicit function theorem to prove it and lemma 3.4 [22] play a great role in this paper. Moreover, when

Main Conclusion
By Hölder inequality, Using the same argument as the process of the proof of Lemma 2.1 in [20] and Hence b m is achieved at b u and Journal of Applied Mathematics and Physics Step 2: Set     Lemma 3.4 [22] and assumption, we get Step 3: For any 0 In fact, by Lemma 2.4,