On multiple solutions to a nonlocal fractional $p(\cdot )$-Laplacian problem with concave–convex nonlinearities

Abstract

The aim of this paper is to examine the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional $p(\cdot )$ p ( ⋅ ) -Laplacian with concave–convex nonlinearities when, in general, the nonlinear term does not satisfy the Ambrosetti–Rabinowitz condition. The main tools for obtaining this result are the mountain pass theorem and a modified version of Ekeland’s variational principle for an energy functional with the compactness condition of the Palais–Smale type, namely the Cerami condition. Also we discuss several existence results of a sequence of infinitely many solutions to our problem. To achieve these results, we employ the fountain theorem and the dual fountain theorem as main tools.

Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions

“…Recently, Di Nezza, Palatucci, and Valdinoci [9] gave a survey on the fractional Sobolev spaces, which are more convenient for fractional Laplacian equations. Lee, Kim, Kim, and Scapellato [18] examined the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional p-Laplacian with concave-convex nonlinearities. Since then, there have been some works on the existence, multiplicity, and concentration phenomenon of solutions to the nonlinear fractional Schrödinger equation (1.2) and other differential problems driven by Laplace-type operators; see [1, 2, 7, 10-13, 15, 19-21, 25, 28-32].…”

Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

“…where V : R N → (0, ∞) is a potential function satisfying (V) and g : R N × R → R fulfills the Carathéodory condition. In particular, in the work [30], the authors obtained the existence of a sequence of small-energy solutions under specific conditions of the nonlinear term that were different from those in previous studies [23,[32][33][34][35][36][37]. More precisely, in view of [32][33][34][35], the conditions of the nonlinear term g near zero as well as at infinity were decisive for proving the hypotheses in the dual-fountain theorem.…”

“…In particular, in the work [30], the authors obtained the existence of a sequence of small-energy solutions under specific conditions of the nonlinear term that were different from those in previous studies [23,[32][33][34][35][36][37]. More precisely, in view of [32][33][34][35], the conditions of the nonlinear term g near zero as well as at infinity were decisive for proving the hypotheses in the dual-fountain theorem. However, the authors also ensured the hypotheses when the behavior at infinity was not assumed, and the condition near zero-namely, g(y, ζ) = o(|ζ| p−2 ζ) as |ζ| → 0 uniformly for all y ∈ R N -was replaced by (G4), which is discussed in Section 2.…”

Multiple weak solutions for pi(x)‐Kirchhoff‐type quasilinear elliptic systems

“…Therefore, weak solutions to these equations are considered. I refer the reader to [1][2][3][4][5][6][7][8][9][10][11] for applications of the p-Laplacian in various contexts, such as game theory, mechanics, and image processing.…”