arXiv subjects

Tuhina Mukherjee

Publications and source records attributed to Tuhina Mukherjee.

40 records · Page 3Linked to original sources

arXiv2016

Positive solutions for nonlinear Choquard equation with singular nonlinearity

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_μ}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_μ-2}u, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{on}\; \partial\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2,\; \la >0,\; 0 < q < 1, \; 0<μ<n$ and $2^*_μ=\frac{2n-μ}{n-2}$. Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if $\la$ is less than some positive constant. We also study the regularity of these weak solutions.

math.AP↗

arXiv2016

Fractional Choquard Equation with Critical Nonlinearities

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ 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.

math.AP↗

arXiv2016

On Dirichlet problem for fractional $p$-Laplacian with singular nonlinearity

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^α, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om. \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > sp, s \in (0,1), \la >0, 0 < q \leq 1 $ and $α\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter $\la$.

math.AP↗

arXiv2016

Critical growth fractional elliptic problem with singular nonlinearities

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-Δ)^s u = λa(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; Ω,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n > 2s,\; s \in (0,1),\; λ>0,\; 0 < q \leq 1 $, $θ\leq a(x) \in L^\infty(Ω)$, for some $θ>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $λ$.

math.AP↗