arXiv subjects

R. Dhanya

Publications and source records attributed to R. Dhanya.

10 recordsLinked to original sources

arXiv2026

Global Multiplicity and Comparison Principles for Singular Problems driven by Mixed Local-Nonlocal Operators

We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_q)^s u &= \fracλ{u^δ} + u^r \text{ in } Ω\newline u > 0 \text{ in } Ω,\ u &= 0 \text{ in } \mathbb{R}^N \setminus Ω \end{aligned} \end{equation*} where $p > sq$, $0<δ<1$ and $λ> 0$ is a parameter. The nonlinearity exhibits a singular power-type behavior near zero and displays at most a critical growth at infinity. We establish a global multiplicity result with respect to the parameter $λ$ by identifying a sharp threshold that separates existence, non-existence, and multiplicity regimes, a result that is new for singular problems involving mixed local-nonlocal operators. We also derive a Hopf-type strong comparison principle adapted to this nonlinear setting, which provides the main analytical tool for the global multiplicity result. Additionally, we investigate qualitative properties of solutions that are essential for the variational analysis, such as a uniform $L^{\infty}$-estimate and a Sobolev versus Hölder local minimizer result. The analytical tools developed herein are of independent mathematical interest, with their applicability extending over a broader class of mixed local-nonlocal problems.

math.AP↗

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

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↗

arXiv2023

Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

We consider the boundary value problem $-Δ_p u_λ-Δ_q u_λ=λg(x) u_λ^{-β}$ in $Ω$ , $u_λ=0$ on $\partial Ω$ with $u_λ>0$ in $Ω.$ We assume $Ω$ is a bounded open set in $\mathbb{R}^N$ with smooth boundary, $1<p<q<\infty$, $β\in [0,1),$ $g$ is a positive weight function and $λ$ is a positive parameter. We derive an estimate for $u_λ$ which describes its exact behavior when the parameter $λ$ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.

math.AP↗

arXiv2023

On a class of infinite semipositone problems for (p,q) Laplace operator

We analyze a non-linear elliptic boundary value problem, that involves $(p, q)$ Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a continuous function in $(0,\infty)$ which is singular, monotonically increasing and eventually positive. We prove the existence of a positive solution of this problem using a fixed point theorem due to Amann\cite{amann1976fixed}. In addition, for a specific nonlinearity we derive that the obtained solution is maximal in nature. The main results obtained here are first of its kind for a $(p, q)$ Laplace operator in an arbitrary bounded domain.

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↗

arXiv2019

A multiparameter semipositone fractional laplacian problem involving critical exponent

In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type $$ (P_λ^μ)\left\{ \begin{array}{lll} (-Δ)^s u&=& λ(u^{q}-1)+μu^r \mbox{ in } Ω\\ u&>&0 \mbox{ in } Ω\\ u&\equiv &0 \mbox{ on }{\mathbb R^N\setminusΩ}. \end{array}\right. $$ when the positive parameters $λ$ and $μ$ belongs to certain range. Here $Ω\subset\mathbb R^N$ is assumed to be a bounded open set with smooth boundary, $s\in (0,1), N> 2s$ and $0 λ_0.$ Now for each $λ>λ_0,$ for all small $0<μ<μ_λ$ we establish the existence of at least one positive solution of $(P_λ^μ)$ using variational method. Also in the subcritical case, i.e., for $1<r<\frac{N+2s}{N-2s}$, we show the existence of second positive solution via mountain pass argument.

math.AP↗

arXiv2016

Elliptic Problems in $\mathbb{R}^N$ with Critical and Singular Discontinuous Nonlinearities

Let $Ω$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, λ>0$ and $0<δ<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $χ_A$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -Δu &= λ\left(u^{2^*-1}+ \displaystyle χ_{\{u 0~~\text{in} ~~Ω, \\ u & = 0 ~\text{on}~ \partial Ω. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to the above problem.

math.AP↗