Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions

Abstract

In this article, we study the existence and multiplicity of non-negative solutions of following p-fractional equation:where Ω is a bounded domain in R n , p ≥ 2, n > pα, α ∈ (0, 1), 0 < q < p − 1 < r < np n−ps − 1, λ > 0 and h, b are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists λ 0 such that for λ ∈ (0, λ 0 ), it has at least two solutions.

On Some Nonlinear Fractional Equations Involving the Bessel Operator

“…More precise references will be found in the next Sections. In particular our results extend those of [2,5,9,20] to our nonlocal model Eq. (1).…”

“…α = 1): see for instance [2][3][4][5]9,16,20]. More precise references will be found in the next Sections.…”

“…In [17] it is shown that (20) has infinitely many solutions. We claim that it possesses a radially symmetric ground state.…”

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

“…Note that the norm involves the interaction between Ω and . This type of functional setting is introduced by Servadei and Valdinoci for in and for in .…”

Multiplicity of Positive Solutions for Critical Fractional Equation Involving Concave–Convex Nonlinearities and Sign-Changing Weight Functions

“…In addition, these operators arise in modelling diffusion and transport in a prevalent role in physical situations such as combustion and highly heterogeneous medium. As to the concave-convex nonlinearity, this type of problems has been studied by many authors [2][3][4][5][6][7] and the references therein. If the weight functions f (x) ≡ g(x) ≡ 1, the authors in [3] have investigated the following equation:…”