On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition

Abstract

The present paper deals with the existence and multiplicity of solutions for a class of fractional p(x, .)-Laplacian problems with the nonlocal Dirichlet boundary data, where the nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. The main results are established by means of mountain pass theorem and Fountain theorem with Cerami condition.

On a Nonlocal Problem Involving Fractional P(X, .)-Laplacian with Non-Standard Growth

“…Next, we present the definition and some results on fractional Sobolev spaces with variable exponent that was introduced in [4,9,19]. Let s be a fixed real number such that 0 < s < 1 and lets the assumptions ( 1) and ( 2) be satisfied, we define the fractional Sobolev space with variable exponent via the Gagliardo approach as follows:…”

“…In [4], the autors study the problem (P). The main results are established by means of mountain pass theorem and Fountain theorem with Cerami condition.…”

Weak solvability of nonlinear elliptic equations involving variable exponents

“…We start by a pioneer work published by Liu and Li in [25] for multiplicity results with superlinear nonlinearities, using the critical point theory with Cerami condition which is weaker than the Palais-Smale condition. For a deeper comprehension, we recommend that readers consult [3,4,6,8,9,10,11] and the references therein. In Sobolev space with variable exponent, some other useful contributions have been devoted to the study but first of all the experts in the field immediately think, among all, to the study made by Colombo and Mingione in [16] and Baroni, Colombo, and Mingione [7].…”