Existence Of Solution And Multiple Solutions For A Class Of Quasilinear Schr?dinger Equation

In this paper, the existence of a positive solution and at least two positive solutions are obtained for a class of quasilinear Schrodinger equation by using the Mountain pass lemma. Ekeland variational principle and the strong maximum prin-ciple. We consider the following problem where N≥3, g(s):Râ†'R+ is a C1 nondecreasing function with respect to |s|, the potential V(x):RNâ†'R is uniformly positive.Firstly, we consider the case m(x)= 0. we suppose that V is a perturbation of periodic function at infinity.The basic tools employed here are the Mountain Pass Theorem and the Con-centration Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation which has an associated functional well defined in the Sobolev space H1(RN) and satisfies the geometry hypotheses of the Mountain Pass Theorem. In the following, we suppose, as a contradiction, that the only possible solution for the problem (P1) is the zero one. Considering the func-tional associated with the problem, we use a version of the Mountain Pass Theorem without the compactness condition to get a Cerami sequence associated with the minimax level. Next, we use this Cerami sequence obtained to get a nontrivial crit-ical point of the functional associated with the periodic problem. Furthermore, we are able to prove that the value of the functional associated with the problem (P1) at this point is less than or equal to the mountain pass minimax level and that this level is attained. Finally, we employ a local version of the Mountain Pass Theorem to obtain a nontrivial critical point of the functional associated with the problem (Pi). This contradicts our initial assumption that the only possible solution for problem (P1) is the zero one.Secondly, we consider the case m(x)(?) 0 and V is a periodic function.By Ekeland variational principle, we show that problem (P1) has a local miniza-tion solution; it follows from the concentration-compactness principle and the Mountain Pass Theorem that problem (P1) has a mountain pass solution. Therefore, problem (P1) possesses at least two positive solutions.