Positive solutions of fractional elliptic equation with critical and singular nonlinearity

Abstract

In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity:(-\Delta)^{s}u=u^{-q}+\lambda u^{{2^{*}_{s}}-1},\qquad u>0\quad\text{in }% \Omega,\qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega,where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega}, {n>2s}, {s\in(0,1)}, {\lambda>0}, {q>0} and {2^{*}_{s}=\frac{2n}{n-2s}}. We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.

Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian

“…We claim that u λ / ∈ X 0 (Ω). On contrary if u λ ∈ X 0 (Ω) then using Lemma 3.1 of [16] and monotone convergence theorem, we can easily show that (3.18) holds for any ψ ∈ X 0 (Ω). Therefore u λ ∈ X 0 (Ω) solves (S) in the weak sense and we get…”

“…More specifically, existence and multiplicity results for the equation (−∆) s u = λu −q + u p in Ω, u = 0 in R n \ Ω has been shown for 0 < q ≤ 1 and 0 < p < 2 * s − 1 where 2 * s = 2n n − 2s in [8] and p = 2 * s − 1 in [19]. Whereas the case q > 0 and p = 2 * s − 1 has been studied in [16]. Concerning the parabolic problems involving the fractional laplacian, we cite [25,26,3,13] and references therein.…”

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

“…We refer the surveys [18] and [24] for further details on singular elliptic equations in the local setting. In the nonlocal case, singular problem with critical nonlinearity has been studied in [6,20,30]. Recently, Adimurthi, Giacomoni and Santra [2] studied the following nonlocal singular problem:…”

A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

“…Many researchers have considerable interest in the existence of solutions and numerical methods for fractional differential equations [2,[6][7][8][9][10][11][12][13], the topics that have been developing rapidly over the last few decades. We are interested in designing an iteration method for solving the spatial Here and in the sequel, we use i = √ -1 to denote the imaginary unit, 1 < α < 2 and γ , ρ > 0, β ≥ 0 are all constants.…”