Existence and multiplicity of solutions for a class of elliptic equations involving nonlocal integrodifferential operator with variable exponent

Abstract

In this work, we are interested in the existence and multiplicity of nontrivial solutions for a class of elliptic problems. The first problem deals with the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator $\mathscr{L}_{\mathcal{A}K}$ with variable exponent, two real parameters, and two weight functions, which can be sign-changing in a smooth bounded domain. Considering different situations related to the growth of nonlinearities involved in problem, we prove the existence of two distinct nontrivial solutions for the case of constant exponents and the existence of a continuous family of eigenvalues in the case of variable exponents. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations.

The second problem deals with the existence and multiplicity of weak solutions involving the same operator $\mathscr{L}_{\mathcal{A}K} $, variable exponents without Ambrosetti and Rabinowitz type growth conditions and a positive real parameter in a smooth bounded domain. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for problem. Moreover, for the case sublinear, by imposing some additional hypotheses on the nonlinearity, we obtain the existence of infinitely many weak solutions which tend to be zero, in the fractional Sobolev norm, for any positive parameter.