Existence of Two Positive Solutions for Two Kinds of Fractional$p$-Laplacian Equations

Abstract

The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator L K p . Specifically, we get multiple solutions to the following fractional p -Laplacian equations with the help of fibering maps and Nehari manifold. − Δ p s u x = λ u q + u r , u > 0 in Ω , u = 0 , in ℝ N \ Ω . . Our results extend the previous results in some respects.

Multiplicity of Solutions for Discrete 2n-TH Order Periodic Boundary Value Problem with φp-Laplacian

“…Refs. [10][11][12][13] is a recent work on this topic that the reader should consult. For instance, the following issue was researched by the authors in [12],…”

Concentration behaviour of normalized ground states of the mass critical fractional Schrödinger equations with ring-shaped potentials

“…|ζ| 2+2s ) −1 (see e.g. [5,17,24] and reference therein). In fact, there are applications of operator (−Δ) s in some areas such as fractional quantum mechanics, physics and chemistry, obstacle problems, optimization and finance, conformal geometry and minimal surfaces, please see [1,2,12,13,15,18] and the references therein for more details.…”

“…It's well known that the fractional Laplacian can be defined by for , where denotes a Cauchy principal value, is the Schwartz space of rapidly decaying function, denotes an open ball of radius centred at and the normalization constant (see e.g. [5, 17, 24] and reference therein). In fact, there are applications of operator in some areas such as fractional quantum mechanics, physics and chemistry, obstacle problems, optimization and finance, conformal geometry and minimal surfaces, please see [1, 2, 12, 13, 15, 18] and the references therein for more details.…”