arXiv subjects

Pawan Kumar Mishra

Publications and source records attributed to Pawan Kumar Mishra.

21 records · Page 2Linked to original sources

arXiv2015

Critical growth fractional systems with exponential nonlinearity

We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-Δ)^{\frac{1}{2}} u &=\frac{p}{p+q}λf(x)|u|^{p-2}u|v|^q + h_1(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1),\\ (-Δ)^{\frac{1}{2}} v &=\frac{q}{p+q}λf(x)|u|^p|v|^{q-2}v + h_2(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1), u,v&>0 \;\textrm{in } \; (-1,1), u&=v=0 \; \text{in} \; \mathbb R\setminus (-1,1). \end{array} \end{equation*} where {$1 2}$. Here $(-Δ)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of multiple solutions for suitable range of $λ$ by analyzing the fibering maps and the corresponding Nehari manifold. We also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity.

math.AP↗

arXiv2015

Existence and multiplicity results for fractional $p$-Kirchhoff equation with sign changing nonlinearities

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-Δ)^{s}_p u &=λf(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} Ω,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus Ω, \end{array} \end{equation*} where $(-Δ)^{s}_p$ is the fractional $p$-Laplace operator, $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(Ω)$ and $g\in L^\infty(Ω)$ are sign changing, $M$ is continuous function, $ps<n<2ps$ and $1<q<p<r\leq p_s^*=\frac{np}{n-ps}$.

math.AP↗

arXiv2015

$n$Kirchhoff type equations with exponential nonlinearities

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_Ω|\nabla u|^n)Δ_n u = f(x,u) \; \text{in}\; Ω,\quad u =0\quad\text{on} \quad \partial Ω, \end{array} \right.$$ where $Ω\subset \mathbf{R}^n$ is a bounded domain with smooth boundary, $n\geq 2$ and $f$ behaves like $e^{|u|^{\frac{n}{n-1}}}$ as $|u|\to\infty$. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when $f(x,t)$ is concave near $t=0$ and convex as $t\rightarrow \infty.$

math.AP↗