Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

Abstract

In this article, we study the following p-fractional Laplacian equationwhere Ω is a bounded domain in R n with smooth boundary, n > pα, p ≥ 2, α ∈ (0, 1), λ > 0 and b : Ω ⊂ R n → R is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (P λ ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p * = np n−pα respectively. We discuss both the cases separately. Non-existence results are also obtained.

A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions

“…As for the fractional case s ∈ (0, 1), we refer to [30] and [27], where variational techniques were used. The use of variational techniques allows for somewhat relaxed hypotheses, however they only give existence of positive solutions of (1.4) for positive values of λ.…”

On doubly nonlocal $p$-fractional coupled elliptic system

“…The Brezis Nirenberg type problem involving p-fractional Laplacian has been studied in [29] whereas existence has been investigated via Morse theory in [30]. Problems involving p-fractional Laplacian has been studied in [24,25] using Nehari manifold. A vast amount of literature can be found for the case p = 2, i.e., fractional Laplacian (−∆) s , which are contributed in recent years.…”

Multiplicity results for the non-homogeneous fractionalp-Kirchhoff equations with concave–convex nonlinearities

“…Here, we first recall the variational framework for problem (1.1), in which most of results can be referred to [42,45]. It is worth mentioning that the functional setting was first introduced by Autuori & Pucci in [46] as p = 2 in R N and Servadei & Valdinoci in [47][48][49] …”