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Akasmika Panda

Publications and source records attributed to Akasmika Panda.

7 recordsLinked to original sources
arXiv2021

A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob} (-\Delta)_p^su&=\mu g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega u&>0,~\text{in}~\Omega, u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega, \end{split} \end{align} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2 0$, $\alpha\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ if $a>0$ and $p_s^*=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent. Under suitable assumptions on the function $g$, we prove the existence of solution to the problem. Furthermore, we show that as $\alpha\rightarrow0^+$, the sequence of solutions of $\eqref{main_prob}$ for each such $\alpha$ converges to a solution of the problem for which $\alpha=0$.

math.AP↗
arXiv2020

A parabolic problem involving $p(x)$-Laplacian, a power and a singular nonlinearity

The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-\Delta_{p(x)}u&=\lambda u^{q(x)-1} + u^{-\delta(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~\Sigma_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~\Omega\nonumber. \end{align*} Here $Q_T=\Omega\times(0,T)$, $\Sigma_T=\partial\Omega\times(0,T)$, $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partial\Omega$, $\lambda\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(\Omega)$, $u_0\in L^r(\Omega)$ with $r\geq 2$, $\delta:\overline{\Omega}\rightarrow(0,\infty)$ is continuous, and $p,q\in C(\overline{\Omega})$ with $\underset{x\in\overline{\Omega}}{\max}~p(x)<N$, $q(\cdot)<p^*(\cdot)$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot)$, $q(\cdot)$.

math.AP↗
arXiv2020

A weighted fractional problem involving a singular nonlinearity and a $L^1$ data

In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-\Delta)_{p,\alpha}^su&= f(x)h(u)+g(x) ~\text{in}~\Omega,\\ u&>0~\text{in}~\Omega,\\ u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega,\nonumber \end{split} \end{equation} where the domain $\Omega\subset \mathbb{R}^N$ is bounded and contains the origin, $ \alpha\in[0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N} 1$ and $h$ is a general singular function with singularity at 0. Further, the fractional $p$-Laplacian with weight $\alpha$ is given by $$(-\Delta)_{p,\alpha}^su(x)=\text{P. V.}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\frac{dy}{|x|^\alpha|y|^{\alpha}},~\forall x\in \mathbb{R}^N.$$

math.AP↗
arXiv2020

Existence of positive solutions for a singular elliptic problem with critical exponent and measure data

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda u^{2_s^*-1}+\mu ~\text{in}~\Omega, u&>0~\text{in}~\Omega, u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega. \end{split} \end{equation} Here, $\Omega$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s<N$, $\lambda,\gamma\in (0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $\mu$ is a nonnegative bounded Radon measure in $\Omega$.

math.AP↗
arXiv2019

Existence results of two mixed boundary value elliptic PDEs in $\mathbb{R}^n$

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $Ω\subset\mathbb{R}^N$ and in $\mathbb{R}^N\setminusΩ$ for $N\geq3$. The boundary $\partialΩ$ of $Ω$ is the decomposition of $Γ_1,Γ_2\subset\partialΩ$ such that $\partialΩ=Γ=\overlineΓ_1\cupΓ_2=Γ_1\cup\overlineΓ_2$ and $Γ_1\capΓ_2=\emptyset$. We have shown that if the Neumann data $f_2\in H^{-\frac{1}{2}}(Γ_2)$ and the Dirichlet data $f_1\in H^{\frac{1}{2}}(Γ_1)$ then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to $f_1\in H^{\frac{1}{2}}(Γ_1)$, $f_2\in H^{-\frac{1}{2}}(Γ_2)$ respectively.

math.AP↗
arXiv2019

A study and an application of the concentration compactness type principle

In this article we develop a concentration compactness type principle in a variable exponent setup. As an application of this principle we discuss a problem involving fractional `{\it $(p(x),p^+)$-Laplacian}' and power nonlinearities with exponents $(p^+)^*$, $p_s^*(x)$ with the assumption that the critical set $\{x\in\Omega:p_s^*(x)=(p^+)^*\}$ is nonempty.

math.AP↗