No abstract

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.

In the present paper, by using the direct variational method and the Ekeland variational principle, we study the existence of solutions for an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition and show the existence of a weak solution.

Abstract:The paper is devoted to the Dirichlet problem for monotone, in general multivalued, elliptic equations with nonstandard growth condition. The growth conditions are more general than the well-known p(x) growth. Moreover, we allow the presence of the so-called Lavrentiev phenomenon. As consequence, at least two types of variational settings of Dirichlet problem are available. We prove results on the existence of solutions in both of these settings. Then we obtain several results on the convergence of certain types of approximate solutions to an exact solution.

Please cite this article in press as: M. Avci, A. Pankov, Nontrivial solutions of discrete nonlinear equations with variable exponent, Abstract. In the present paper, we show the existence of ground state solution of a discrete p(n)-Laplacian type equation involving unbounded potential by using the Mountain-Pass theorem and Nehari manifold.