Existence of Ordered Solutions to Quasilinear Schrödinger Equations with General Nonlinear Term

In this paper, the existence of a pair of ordered solutions for the following class of equations in (1) was studied. A bounded (PS) (Palais-Smale) sequence was constructed and the related variational principle was used to prove the existence of the positive solution. The existence of the ordered solutions is finally


Introduction
In recent years, studies about the nontrivial solutions of Schrödinger equations are very popular, involving differential equations, linear algebra and many subjects. The solution of these problems cannot only develop new methods, such as minimizations [1] [2], change of variables [3] [4] [5], Nehari method [6] and perturbation method [7], reveal new laws, but also have important academic value and wide application prospects [8] [9].
In this paper, we consider the existence of ordered solutions for the following quasilinear Schrödinger equations: we make the following assumptions:  , , (g 6 ) There exists * 0 C > , such that

Main Results
We are now state the main results of the paper:

Preliminaries
We observe that formally problem (2) is the Euler-Lagrange equation associated of the natural energy functional given by It is well known that J is not well defined in general in ( ) which is well defined in the space We can see that the nontrivial critical points of ( ) Lemma 3.1 (see in [4]) The function ( ) f t possesses the following properties.

J. Wu Journal of Applied Mathematics and Physics
Finally we prove (9). For any 0 t > , we have the following inequality by (5) For all t ∈  , we have 2) For each λ ς ∈ and for all v X 3) There exist two points 1 2 where [ ] In order to use Lemma 3.2, in the following discussion, we take and consider the following family of functional The following lemma shows that Lemma 3.3 Assume conditions (V 1 ) and (g 1 )-(g 4 ) are satisfied, we have Proof: (1) can be directly obtained from (g 4 ). Let's prove (2) by Lemma 3.1 and embedding theorem, we infer that Therefore, A is convex.

J. Wu Journal of Applied Mathematics and Physics
To prove (3), firstly, we let Then fixing a non-negative radial symmetry function Thus there exists ( ) Finally, we prove (4). Define and Lemma 3.1, we have Hence, there exists 0 C > , such that ( ) It follows that By (i) and (ii), we obtain By (g 1 ), (g 2 ) and Lemma 3.1, there exists 1 2 , 0 C C > , for every N x ∈  and for all t ∈  , such that By (8) and  Similarly, let Proof: Multiply the two sides of the equation Finally, we integrate the equation on N  , and then the improved Pohozaev type identity can be obtained.  Thus, for any 0

Existence Results
Proof of Theorem 1.
is the non-negative of Equation (2), so that ( ), 0 By This implies that inequality (11) is satisfied. This completes the proof. (2) admits a positive solution v, and v is a local minimizer for ( ) Proof: According to the reference [10] and related theories of differential equations, Equation (2)  The following defines some sets and functions: And then ( ) Obviously, n v u = on n S , so that ( ) ( Since u is a sub-solution, we obtain Similarly, u is a sup-solution, so that Hence, In addition, we noticed that ( ) To complete the Lemma, we still need to prove the following claim: as n → ∞ , Since the proofs of inequalities (15) and (16) are similar, we only prove (15) Firstly, note By the define of f, for any t ∈  , we have By differential mean value theorem, Lemma 3.1 (1) and (17), we have Moreover, by the define of n S , we have lim 0 n n S →∞ = . In fact, for any 0 Again since , , It follows from differential mean value theorem that ( )