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Ratan Kr. Giri

Publications and source records attributed to Ratan Kr. Giri.

8 recordsLinked to original sources
arXiv2026

Optimization for the first weighted eigenvalue of local-nonlocal operators with a potential

In this work, we study the optimization problem for the first eigenvalue of mixed local nonlocal operators plus a potential $V$. We begin by investigating the existence and fundamental properties of the eigenvalues, with special emphasis on the first eigenvalue. Finally, we discuss the dependence of the first eigenvalue on the potential function and establish the existence of optimal potentials within certain admissible classes. In particular, we show the existence of a unique maximizer and a minimizer of the first eigenvalue on any bounded, closed, and convex subset of $L^q(Ω)$. Moreover, these results enable us to characterize the maximizers and minimizers in the closed unit ball of $L^q(Ω)$, as well as in the class of rearrangements of any $V\in L^q(Ω)$.

math.AP↗
arXiv2024

A weighted eigenvalue problem for mixed local and nonlocal operators with potential

We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $Ω\subset \mathbb{R}^N \,(N\geq2)$ with {\it Lipschitz boundary} $\partial Ω$, which is given by \begin{align*} -Δ_p u + (-Δ_p)^su+V(x)|u|^{p-2}u&=λg(x)|u|^{p-2}u~\text{in}~Ω, u&=0~\text{in}~\mathbb{R}^N\setminusΩ, \end{align*} where $λ>0$ is a parameter, exponents $0 0$ a.e. in $Ω$. Using the variational tools together with a {\it weak comparison} and {\it strong maximum principles}, we investigate the existence and uniqueness of {\it principal eigenvalue} and discuss its qualitative properties. Moreover, with the help of {\it Ljusternik-Schnirelman category theory}, it is proved that there exists a {\it nondecreasing sequence of positive eigenvalues} which goes to infinity. Further, we show that {\it the set of all positive eigenvalues is closed}, and {\it eigenfunctions} associated with every {\it positive eigenvalue} are bounded.

math.AP↗
arXiv2022

Positive solutions of quasilinear elliptic equations with Fuchsian potentials in Wolff class

Using Harnack's inequality and a scaling argument we study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in \partialΩ\cup\{\infty\}$ for the quasilinear elliptic equation $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{ in } Ω,$$ where $Ω$ is a domain in $\mathbb{R}^d$, $d\geq 2$, $1<p<d$, and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω; \mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. It is assumed that the potential $V$ belongs to a certain Wolff class and has a generalized Fuchsian-type singularity at an isolated point $ζ\in \partial Ω\cup \{\infty\}$.

math.AP↗
arXiv2021

Supercaloric functions for the parabolic $p$-Laplace equation in the fast diffusion case

We study a generalized class of supersolutions, so-called $p$-supercaloric functions, to the parabolic $p$-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $p\geq 2$, but little is known in the fast diffusion case $1<p<2$. Every bounded $p$-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic $p$-Laplace equation for the entire range $1<p<\infty$. Our main result shows that unbounded $p$-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $\frac{2n}{n+1}<p<2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $1<p\leq \frac{2n}{n+1}$ and the theory is not yet well understood.

math.AP↗
arXiv2020

Positive Liouville theorem and asymptotic behaviour for $(p,A)$-Laplacian type elliptic equations with Fuchsian potentials in Morrey space

We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in\partialΩ\cup\{\infty\}$ of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } Ω\setminus\{ζ\},$$ where $Ω$ is a domain in $\mathbb{R}^d$ ($d\geq 2$), and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω;\mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $ζ$.

math.AP↗
arXiv2020

Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-Δ)_{p(\cdot)}^{s} u&=\fracλ{|u|^{γ(x)-1}u}+f(x,u)~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω\subset\mathbb{R}^N,\, N\geq2$ is a smooth, bounded domain, $λ>0$, $s\in (0,1)$, $γ(x)\in(0,1)$ for all $x\in\barΩ$, $N>sp(x,y)$ for all $(x,y)\in\barΩ\times\barΩ$ and $(-Δ)_{p(\cdot)}^{s}$ is the fractional $p(\cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{\infty}(\barΩ)$ estimate of the solution(s) by the Moser iteration technique.

math.AP↗
arXiv2017

Problem involving nonlocal operator

The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional $p$-Laplacian operator. We prove the existence of a solution in the weak sense to the problem \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u\,\,\mbox{in}\,\,Ω,\\ u & = 0\,\, \mbox{in}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} if and only if a weak solution to \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +f,\,\,\,f\in L^{p'}(Ω),\\ u & = 0\,\, \mbox{on}\,\, \mathbb{R}^N\setminus Ω\end{split} \end{align*} ($p'$ being the conjugate of $p$), exists in a weak sense, for $q\in(p, p_s^*)$ under certain condition on $λ$, where $-\mathscr{L}_Φ$ is a general nonlocal integrodifferential operator of order $s\in(0,1)$ and $p_s^*$ is the fractional Sobolev conjugate of $p$. We further prove the existence of a measure $μ^{*}$ corresponding to which a weak solution exists to the problem \begin{align*} \begin{split} -\mathscr{L}_Φu & = λ|u|^{q-2}u +μ^*\,\,\,\mbox{in}\,\, Ω,\\ u & = 0\,\,\, \mbox{in}\,\,\mathbb{R}^N\setminus Ω\end{split} \end{align*} depending upon the capacity.

math.AP↗