arXiv subjects

Sarbani Pramanik

Publications and source records attributed to Sarbani Pramanik.

3 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↗

arXiv2026

Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_p)^s u = λ\frac{f(u)}{u^β}\ \text{in} \ Ω\newline u >0\ \text{in} \ Ω,\ u =0\ \text{in} \ \mathbb{R}^N \setminus Ω\end{aligned} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $λ> 0 $ is a parameter, $0\leq β<1$ and $f$ is a non-decreasing $C^1$-function with $f(0)>0$. Using the method of sub- and supersolutions, we present a novel multiplicity result and, in specific cases, we also prove a three-solution theorem using Amann's fixed point theorem. Our construction of sub-supersolutions avoids the conventional reliance on ODE techniques and Green's function estimates, thereby making it more adaptable to the nonlinear and nonlocal framework. Additionally, we establish a Hopf-type Strong Comparison Principle for the linear operator with singular nonlinearity, marking the first result of its kind for mixed local-nonlocal operators. This result is crucial in deriving a third solution and holds broader mathematical significance.

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↗