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Prashanta Garain

Publications and source records attributed to Prashanta Garain.

47 records · Page 3Linked to original sources
arXiv2021

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.

math.AP↗
arXiv2021

Weak Harnack inequality for a mixed local and nonlocal parabolic equation

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.

math.AP↗
arXiv2021

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case and in the nonlocal elliptic case to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

math.AP↗
arXiv2020

On a degenerate singular elliptic problem

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u^δ}\,\,\text{ in }\,\,Ω, u>0\text{ in }Ω,\\ u = 0 \text{ on } \partialΩ, \end{cases} \end{gather*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 2$, $w$ belong to the Muckenhoupt class $A_p$ for some $1 0$.

math.AP↗
arXiv2020

Existence and Nonexistence results for anisotropic p-Laplace equation with singular nonlinearities

Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }Ω. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\frac{1}{u}$ in a bounded smooth domain $Ω$ and nonexistence results for $f(u)=-e^\frac{1}{u}$ or $-(u^{-δ}+u^{-γ}),$ $δ,γ>0$ with $Ω=\mathbb{R}^N$ respectively.

math.AP↗
arXiv2020

Some local properties of subsolutons and supersolutions for a doubly nonlinear nonlocal parabolic $p$-Laplace equation

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.

math.AP↗
arXiv2018

Properties of solutions to some weighted $p$-Laplacian equation

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$.

math.AP↗
arXiv2017

Non-existence of stable solutions for weighted $p$-Laplace equation

We provide sufficient conditions on $w\in L^1_{loc}(\mathbb{R}^N)$ such that the weighted $p$-Laplace equation $$-\operatorname{div}\big(w(x)|\nabla u|^{p-2}\nabla u\big)=f(u)\;\;\mbox{in}\;\;\mathbb{R}^N$$ does not admit any stable $C^{1,ζ}_{loc}$ solution in $\mathbb{R}^N$ where $f(x)$ is either $-x^{-δ}$ or $e^x$ for any $0<ζ<1$.

math.AP↗
arXiv2017

Multiplicity results for a quasilinear equation with singular nonlinearity

For an open, bounded domain $\Om$ in $\mathbb{R}^N$ which is strictly convex with $C^2$ boundary, we show that there exists a $\land>0$ such that the singular quasilinear problem \begin{eqnarray*} &-\delp u =\cfracλ{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atleast two solution $ u$ and $v$ in $W^{1,p}_{loc}(\Om)\cap L^{\infty}(\Om)$ for any $\del>0$ and $0<\lam<\land$ provided $1 0$.

math.AP↗