Tuhina Mukherjee

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: ( - Δ ) s ⁢ u = u - q + λ ⁢ u 2 s * - 1 , u > 0   in ⁢ Ω , u = 0   in ⁢ ℝ n ∖ Ω , (-\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 ℝ n {\mathbb{R}^{n}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} , n > 2 ⁢ s {n>2s} , s ∈ ( 0 , 1 ) {s\in(0,1)} , λ > 0 {\lambda>0} , q > 0 {q>0} and 2 s * = 2 ⁢ n n - 2 ⁢ s {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.

Fractional Choquard equation with critical nonlinearities

In this article, we study the Brezis–Nirenberg type problem of nonlinear Choquard equation involving the fractional Laplacian

Positive solutions for nonlinear Choquard equation with singular nonlinearity

\begin{aligned} (-\Delta )^s u = \left( \int _{\Omega }\frac{|u|^{2^*_{\mu ,s}}}{|x-y|^{\mu }}\mathrm {d}y \right) |u|^{2^*_{\mu ,s}-2}u +\lambda u \; \text {in } \Omega , \quad {u=0 \; \text {in}\; \mathbb R^n{\setminus }\Omega }, \end{aligned}

p

(-Δ)su=∫Ω|u|2μ,s∗|x-y|μdy|u|2μ,s∗-2u+λuinΩ,u=0inRn\Ω,where

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

\Omega

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

Ω is a bounded domain in