Existence Of Positive Solutions For P(x)-Laplacian Equations In Unbounded Domains

In this paper we are concerned with the existence of weak solutions of the followingp(x)-Laplace equations under some conditions.Because of the lack of compactness for the Sobolev space on unbounded domain and then (PS)-condition fails, the above problem become more complex. To overcome this difficulty, we prove the compactness of concentration, and analyze (PS)-sequence, and use their convergence to ensure the existence of critical points for the correspondingfunctional. The tools we used here are Ekeland Variational Principle, Nehari manifold, compactness of concentration, imbedding theory of generalized Lebesgue-Sobolev spaces and so on. The existence is affected by the properties of the geometry and the topology of the domain. The problem has a weak solution, has one ground state solution, have a ground state solution and a higher energy solution respectively when the domain satisfy different conditions .