EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR EQUATIONS OF p(x)-LAPLACE TYPE IN R-N WITHOUT AR-CONDITION

Abstract

We are concerned with the following elliptic equations with variable exponents -div(phi(x, del u)) + V(x)vertical bar u vertical bar(p(x)-2)u = lambda f(x, u) in R-N, where the function phi(x, v) is of type vertical bar v vertical bar(p(x)-2)v with continuous function p : R-N -> (1, infinity), V : R-N -> (0, infinity) is a continuous potential function, and f : R-N x R -> R satisfies a Caratheodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass theorem and fountain theorem. Second, we determine the precise positive interval of lambda's for which our problem admits a nontrivial solution with simple assumptions in some sense.