In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacianwhere Ω is a bounded domain in R n with Lipschitz boundary, λ is a real parameter, s ∈ (0, 1), n > 2s and 2 * µ,s = (2n − µ)/(n − 2s) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.

This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see (KC) below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to (KC). Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem (P λ,M ) below.

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.

In this article, we study the following fractional p-Laplacian equation with critical growth singular nonlinearitywhere Ω is a bounded domain in R n with smooth boundary ∂Ω, n > sp, s ∈ (0, 1), λ > 0, 0 < q ≤ 1 and α ≤ p * s − 1. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter λ.

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity