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Reshmi Biswas

Publications and source records attributed to Reshmi Biswas.

14 recordsLinked to original sources
arXiv2026

Fully Nonlinear Elliptic Grad--Mercier Equations in Weighted Orlicz Spaces

In this article, we study the existence and global regularity results for the fully nonlinear elliptic Grad--Mercier type equations with oblique boundary conditions in the context of weighted Orlicz spaces. Our approach employs an asymptotic analysis in which global regularity is transferred from a limit profile, namely, the recession operator associated with the governing operator, using topological and stability methods. In addition to the main regularity result, we derive global weighted Orlicz estimates for the Hessian and establish global Morrey-type estimates for the problem. This article extends the results established by Caffarelli--Tomasetti (Comm. Pure Appl. Math. 76 (3): 604--615, 2023), Zhang et al. (Nonlinearity 39 (2): 025011, 2026), and Bessa (J. Funct. Anal. 286 (4): 110295, 2024).

math.AP↗
arXiv2025

Comparison Principle, A.B.P.-type estimates for solutions of quasi-linear elliptic equations in non-divergence form and some implications

In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$ -\Psi(x, |\nabla u|)\Delta_p^{\mathrm{N}}u(x)+\mathscr{H}(x,\nabla u)=f(x) \quad \mathrm{in} \quad \Omega, \quad \mathrm{for} \,\,\,1<p< \infty, $$ under suitable assumptions on the data of the problem. Particularly, our results are relevant for a class of quasi-linear models with Hamiltonian terms. Additionally, we address non-degeneracy estimates for such solutions and present a couple of applications.

math.AP↗
arXiv2024

Nonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains

This article is concerned with the nonexistence of positive solutions for nonlinear fractional elliptic inequalities in exterior domains of $\mathbb R^n$, $n\geq1$. We would like to highlight the fact that our results are new with very weak assumption on the nonlinear term appearing in the inequalities. The results are also novel in this generality even if the whole space is considered.

math.AP↗
arXiv2023

Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schrödinger} equations involving Stein-Weiss type convolution \begin{align*} -Δ_N u -Δ_N (u^{2})u +V(x)|u|^{N-2}u= \left(\int_{\mathbb R^N}\frac{F(y,u)}{|y|^β|x-y|^μ}~dy\right)\frac{f(x,u)}{|x|^β} \;\; \text{ in}\; \mathbb R^N, \end{align*} where $N\geq 2,\,$ $0<μ<N,\, β\geq 0,$ and $2β+μ\leq N.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous function satisfying $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some appropriate assumptions. The nonlinearity $f:\mathbb R^N\times \mathbb R\to \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int_{0}^s f(x,t)dt$ is the primitive of $f$.

math.AP↗
arXiv2021

Regularity results for Choquard equations involving fractional $p$-Laplacian

In this article, first we address the regularity of weak solution for a class of $p$-fractional Choquard equations: \begin{equation*} \;\;\; \left.\begin{array}{rl} (-Δ)_p^su&=\left(\displaystyle\int_Ω\frac{F(y,u)}{|x-y|^μ}dy\right)f(x,u),\hspace{5mm}x\in Ω, u&=0,\hspace{35mm}x\in \mathbb R^N\setminus Ω, \end{array} \right\} \end{equation*} where $Ω\subset\mathbb R^N$ is a smooth bounded domain, $1<p<\infty$ and $0<s<1$ such that $sp<N,$ $0<μ<\min\{N,2sp\}$ and $f:Ω\times\mathbb R\to\mathbb R$ is a continuous function with at most critical growth condition (in the sense of Hardy-Littlewood-Sobolev inequality) and $F$ is its primitive. Next, for $p\geq2,$ we discuss the Sobolev versus Hölder minimizers of the energy functional $J$ associated to the above problem, and using that we establish the existence of the local minimizer of $J$ in the fractional Sobolev space $W_0^{s,p}(Ω).$ Moreover, we discuss the aforementioned results by adding a local perturbation term (at most critical in the sense of Sobolev inequality) in the right-hand side in the above equation.

math.AP↗
arXiv2020

Multiplicity and uniform estimate for a class of variable order fractional $p(x)$-Laplacian problems with concave-convex nonlinearities

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+\left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is continuous function with $F(x,t):=\displaystyle\int_{0}^{t} f(x,s)ds$. Under suitable assumption on $s,p,μ,α$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

math.AP↗
arXiv2020

On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)-$ Laplacian and without Ambrosetti-Rabinowitz type condition

In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)-$ Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.

math.AP↗
arXiv2020

Nehari manifold for fractional p(.)-Laplacian system involving concave-convex nonlinearities

In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \begin{equation*} \;\;\; \begin{array}{rl} (-Δ)_{p(\cdot)}^{s} u&=λ~ a(x)| u|^{q(x)-2}u+\frac{α(x)}{α(x)+β(x)}c(x)| u|^{α(x)-2}u| v| ^{β(x)},\hspace{2mm} x\in Ω; \\ (-Δ)_{p(\cdot)}^{s} v&=μ~ b(x)| v|^{q(x)-2}v+\frac{α(x)}{α(x)+β(x)}c(x)| v|^{α(x)-2}v| u| ^{β(x)},\hspace{2.5mm} x\in Ω; \\ u=v&=0 ,\hspace{1cm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $Ω\subset\mathbb R^N,~N\geq2$ is a smooth bounded domain, $λ,μ>0$ are the parameters, $s\in(0,1),$ $p\in C(\mathbb R^N\times \mathbb R^N,(1,\infty))$ and $q,α,β\in C(\overlineΩ,(1,\infty))$ are the variable exponents and $a,b,c\in C(\overlineΩ,[0,\infty))$ are the non-negative weight functions. We show that there exists $Λ>0$ such that for all $λ+μ<Λ$, there exist two non-trivial and non-negative solutions of the above problem under some assumptions on $q,α,β$.

math.AP↗
arXiv2019

Variable order nonlocal Choquard problem with variable exponents

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+ \left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)), x\in Ω, u(x)&=0, x\in \mathbb R^N\setminusΩ, where $Ω\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carathédory function. Under suitable assumption on $s,p,μ,α$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

math.AP↗