A Global multiplicity result for a very singular critical nonlocal equation

Abstract

In this article, we show the global multiplicity result for the following nonlocal singular problemwhere Ω is a bounded domain in R n with smooth boundary ∂Ω, n > 2s, s ∈ (0, 1), λ > 0, q > 0 satisfies q(2s − 1) < (2s + 1) and 2 * s = 2n n−2s . Employing the variational method, we show the existence of at least two distinct weak positive solutions for (P λ ) in X 0 when λ ∈ (0, Λ) and no solution when λ > Λ, where Λ > 0 is appropriately chosen. We also prove a result of independent interest that any weak solution to (P λ ) is in C α (R n ) with α = α(s, q) ∈ (0, 1). The asymptotic behaviour of weak solutions reveals that this result is sharp.

Regularity results on a class of doubly nonlocal problems

“…where q > 0 and the function f is of subcritical growth. When f has critical growth then the question of existence and regularity have been answered in [18].…”

Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities

“…Here authors extended the techniques of [29] in fractional framework and proved the existence and multiplicity of solutions in C α loc (Ω) ∩ L ∞ (Ω) for some α > 0. Recently, authors [20] proved the global multiplicity result for (1.3) with a = 1/λ, p = 2 * s − 1 and r(2s − 1) < (1 + 2s) for energy solutions. Concerning the doubly nonlocal problem with singular operators, in [19], we studied the regularity results for the problems of th type (P λ ) with 0 < q < 1.…”

Fractional Schrödinger–poisson System With Singularity: Existence, Uniqueness, and Asymptotic Behavior

“…Fiscella [5,6] proved the existence of two solutions for problem (1.2) involving Kirchhoff prototype and a critical nonlinearity. Giacomoni et al [9,10] investigated the existence, multiplicity, and Hölder regularity of weak solutions for problem (1.2) with critical growth for any γ > 0. Finally, we also mention [1,11] where the existence of three solutions and bifurcation results were given for different fractional singular problems.…”