arXiv subjects

Ritabrata Jana

Publications and source records attributed to Ritabrata Jana.

7 recordsLinked to original sources

arXiv2025

Fine Boundary Regularity For The Fractional (p,q)-Laplacian

In this article, we deal with the fine boundary regularity, a weighted Hölder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-Δ)_{p}^{s} u + (-Δ)_{q}^{s} u = f(x)$ in $Ω,$ and $u=0$ in $\mathbb{R}^N\setminusΩ;$ where $Ω$ is a $C^{1,1}$ bounded domain and $2 \leq p \leq q <\infty.$ For $0<s<1$ and for non-negative data $f\in L^{\infty}(Ω),$ we employ the nonlocal analogue of the boundary Harnack method to establish that $u/{d_Ω^{s}} \in C^α(\BarΩ)$ for some $α\in (0,1),$ where $d_Ω(x)$ is the distance of $x$ from the boundary. A novel barrier construction allows us to analyse the regularity theory even in the absence of the scaling or the homogeneity properties of the operator. Additionally, we extend our idea to sign changing bounded $f$ as well and prove a fine boundary regularity for fractional $(p,q)$ Laplacian for some range of $s.$

math.AP↗

arXiv2025

Multiplicity Results for Mixed Local Nonlocal Equations With Indefinite Concave-Convex Type Nonlinearity

In this article we examine the multiplicity of non-negative solutions to mixed local-nonlocal equations involving $(-Δ_p) + (-Δ^{s}_{q})$ in a bounded smooth domain. The nonlinearity incorporates a parameter $λ> 0$, a sublinear term, and a superlinear term, with sign-changing weight functions $a(x)$ and $b(x)$. Under suitable conditions, we establish the existence of at least two distinct nontrivial non-negative solutions in both the subcritical and critical regimes via fibering map analysis and constrained minimization on the Nehari manifold. Additionally, for $p \not = q$, we obtain a nonexistence result for large $λ$ by analyzing the associated generalized eigenvalue problem.

math.AP↗

arXiv2024

Interior and Boundary Regularity of Mixed Local Nonlocal Problem with Singular Data and Its Applications

In this article, we examine the Hölder regularity of solutions to equations involving a mixed local-nonlocal nonlinear nonhomogeneous operator $\fp + \fqs$ with singular data, under the minimal assumption that $p> sq$. The regularity result is twofold: we establish interior gradient Hölder regularity for locally bounded data and boundary regularity for singular data. We prove both boundary Hölder and boundary gradient Hölder regularity depending on the degree of singularity. Additionally, we establish a strong comparison principle for this class of problems, which holds independent significance. As the applications of these qualitative results, we further study sublinear and subcritical perturbations of singular nonlinearity.

math.AP↗

arXiv2023

Symmetry and Monotonicity Property of a Solution of (p,q) Laplace Equation with Singular Term

This paper examines the behavior of a positive solution $u\in C^{1,α}(\BarΩ)$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation: \begin{equation*} -div(|\nabla u|^{p-2}\nabla u+ a(x) |\nabla u|^{q-2}\nabla u) &= \frac{g(x)}{u^δ}+h(x)f(u) \, &\text{in} \thinspace B_R(x_0), \quad u & =0 \ &\text{on} \ \partial B_R(x_0). \end{equation*} We assume that $0<δ<1$, $1<p\leq q<\infty$, and $f$ is a $C^1(\mathbb{R})$ nondecreasing function. Our analysis uses the moving plane method to investigate the symmetry and monotonicity properties of $u$. Additionally, we establish a strong comparison principle for solutions of the $(p,q)$ Laplace equation with radial symmetry under the assumptions that $1<p\leq q\leq 2$ and $f\equiv1$.

math.AP↗

arXiv2023

Positive Solutions for Fractional p- Laplace Semipositone Problem with Superlinear Growth

We consider a semipositone problem involving the fractional $p$ Laplace operator of the form \begin{equation*} \begin{aligned} (-Δ)_p^s u &=μ( u^{r}-1) \text{ in } Ω,\\ u &>0 \text{ in }Ω,\\ u &=0 \text{ on }Ω^{c}, \end{aligned} \end{equation*} where $Ω$ is a smooth bounded convex domain in $\mathbb{R}^N$, $p-1<r<p^{*}_{s}-1$, where $p_s^{*}:=\frac{Np}{N-ps}$, and $μ$ is a positive parameter. We study the behaviour of the barrier function under the fractional $p$-Laplacian and use this information to prove the existence of a positive solution for small $μ$ using degree theory. Additionally, the paper explores the existence of a ground state positive solution for a multiparameter semipositone problem with critical growth using variational arguments.

math.AP↗

arXiv2022

Strong comparison principle for a p-Laplace equation involving singularity and its applications

In this paper we prove a strong comparison principle for radially decreasing solutions $u,v\in C_{0}^{1,α}(\Bar{B_R})$ of the singular equations $-Δ_p u-\frac{1}{u^δ}=f(x)$ and $-Δ_p v-\frac{1}{v^δ}=g(x)$ in $B_R$. Here we assume that $ 1 2$ a counterexample is provided where the strong comparison principle is violated. As an application of strong comparison principle, we prove a three solution theorem for p-Laplace equation and illustrate with an example.

math.AP↗

arXiv2021

Proof of a Conjecture on Wiener Index and Eccentricity of a graph due to edge contraction

For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by $\varepsilon(G)$, is the sum of the eccentricity of individual vertices. In \cite{Kc}, the authors posed a conjecture which states that given a graph $G$ with at least three vertices, the difference between $W(G)$ and $\varepsilon(G)$ decreases when an edge is contracted and proved that the conjecture is true when $e$ is a bridge. In this manuscript, we confirm that the conjecture is true for any connected graph $G$ with at least three vertices irrespective of the nature of the edge chosen.

math.CO↗