Sign changing solutions of p-fractional equations with concave-convex nonlinearities

Abstract

In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities:where s ∈ (0, 1) and p ≥ 2 are fixed parameters, 0 < q < p − 1, µ ∈ R + and p * s = Np N−ps . Ω is an open, bounded domain in R N with smooth boundary with N > ps .2010 Mathematics Subject Classification. 47G20, 35J20, 35J60, 35J62.

Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

“…Recall that any critical point u of I λ,p (u) is a weak solution for (1). The starting point of the study of existence of weak solutions to problem (1) is therefore the following fractional inequalities which will guarantee that the above functional is well defined and bounded below on the right function spaces.…”

“…Addressing the questions regarding the effect of concave-convex non-linearities on the number of positive solutions for non-local elliptic problems has been the subject of several studies; see [1][2][3][4] and [17]. Barrios-Medina-Peral [4] studied the sub-critical case of (1), and proved that there exists Λ > 0 such that the problem has at least two solutions for all 0 < λ < Λ when f (x) = g(x) ≡ 1, s = 0, and γ < γ H (α).…”

“…Recently, Zhang-Liu-Jiao [25] extended the results of [2] to the problem involving the sign-changing weight function f (x) ∈ C(Ω) with f + ≡ 0, and g(x) ≡ 1. When α = 2, i.e., in the case of the standard Laplacian, problem (1) has been studied extensively in the last decade; see for example [18], [19], [20], [24] and references therein. We point out that the non-local problems are still much less understood than their local counterpart.…”

“…The aim of this paper is to consider the remaining cases and generalize the results of [4] and [25] to the problem involving the Hardy potential, Hardy-Sobolev singularity term, and also signchanging functions f (x) and g(x). More pricesly, we study problem (1) in the sub-critical (i.e., when 2 < p < 2 * α (s)) and critical (i.e., when p = 2 * α (s)) case, separately. Using the decomposition of the Nehari manifold as λ varies, introduced by Tarantello [24], we will prove that the problem has at least two positive solutions for λ sufficiently small.…”

Multiplicity results for non-local elliptic problems with jumping nonlinearity

“…The existence of sign-changing solutions for p = 2 has been studied in [37]. In the non-local case, multiplicity results and existence of sign-changing solutions have been discussed by several authors, we refer a few among them (see [5], [3], [4], [20] and the refernces therein). Very recently, in [5], the authors have established the existence of infinitely many nontrivial solutions for the class of (p, q) fractional elliptic equations in bounded domains in R N .…”