Theorem 3.7Let J 2 C 1 .X, R/ be functional satisfying the .PS/ condition. Furthermore, let us suppose that (i) J is bounded from below and even; (ii) There is a compact set K 2 R such that .K/ D k and sup x2K J .x/ < J .0/.Then J possesses at least k pairs of distinct critical points, and their corresponding critical values are less than J .0/. Lemma 3.8Suppose .M 1 /, .f 1 /, and q C <ˇp hold. Then I is bounded from below.

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This paper deals with the existence of solutions for some elliptic equations with nonstandard growth under zero Dirichlet boundary condition. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions.

We show the existence of a week solution in "Equation missing" to a Dirichlet problem for "Equation missing" in "Equation missing", and its localization. This approach is based on the nonlinear alternative of Leray-Schauder.