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.
In this paper, using the Nehari manifold approach and some variational techniques, we discuss the multiplicity of positive solutions for the p(x)-Laplacian problems with non-negative weight functions and prove that an elliptic equation has at least two positive solutions.
Keywords:Variable exponent Lebesgue and Sobolev spaces Nonlinear parabolic variational inequality In this article, new properties of variable exponent Lebesgue and Sobolev spaces are examined. Using these properties we prove the existence of the solution of some parabolic variational inequality.
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