arXiv subjects

Sanjit Biswas

Publications and source records attributed to Sanjit Biswas.

8 recordsLinked to original sources

arXiv2026

On the regularity theory for mixed local and nonlocal weighted quasilinear elliptic equations

We investigate a broad class of mixed local and nonlocal degenerate $p$-Laplace equations with general right-hand sides. The degeneracy is governed by Muckenhoupt $A_p$-weights, yielding a highly nonuniform elliptic framework in which both the local and nonlocal operators may degenerate simultaneously. We establish a comprehensive local regularity theory, including local boundedness and lower semicontinuity of weak subsolutions, weak Harnack inequalities for weak supersolutions, Harnack inequalities, and local H\"older continuity of weak solutions. Our approach combines weighted analytic techniques with the De Giorgi--Nash--Moser iteration method, adapted to the mixed local--nonlocal setting. To the best of our knowledge, this is the first systematic regularity theory for mixed local and nonlocal equations with Muckenhoupt weights. In particular, our results are new even for homogeneous linear equations ($p=2$) under the natural assumption $w\in A_2$, and therefore substantially extend the existing regularity theory for mixed local--nonlocal equations to a degenerate weighted framework with general right-hand sides.

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arXiv2026

Symmetry and Approximate Symmetry for Solutions of Mixed Local-Nonlocal Singular Equations

In this article, we establish radial symmetry for positive weak solutions of a class of mixed local-nonlocal equations with possibly singular nonlinearity via the moving plane method. Furthermore, we provide a quantitative version of Gidas-Ni-Nirenberg type theorem for mixed local-nonlocal equations. In this regard, we establish a weak Harnack-type inequality and an analogue of the Alexandroff-Bakelman-Pucci inequality in the mixed nonhomogeneous setting with a lower order term, which appear to be new. To the best of our knowledge, this paper initiates the study of the quantitative properties of solutions to mixed problems.

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arXiv2025

Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures

This article establishes the existence of weak solutions for a class of mixed local-nonlocal problems with pure and perturbed singular nonlinearities. A key novelty is the treatment of variable singular exponents alongside measure-valued data. Notably, both source terms may be measures, with the singular component modeled by both a singular and non-singular measure. Our main focus is on the singular measure data, which appears to be new, even for constant exponents.

math.AP↗

arXiv2025

Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2

In this article, we prove the existence of at least two positive weak solutions for a mixed local-nonlocal singular problem in the presence of critical exponential nonlinearity in dimension two. The novelty of this work is the inclusion of a variable singular exponent in the context of mixed operator and critical exponential nonlinearity in R^2. Our approach is based on sub-solution super-solution technique, combined with variational methods.

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arXiv2025

Regularity and existence for semilinear mixed local-nonlocal equations with variable singularities and measure data

This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another notable feature is that the source terms in both the purely singular and perturbed components can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent.

math.AP↗

arXiv2024

Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases

Given a smooth, bounded domain $\Omega\subset\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \begin{align*} \left. \begin{array}{l} -\Delta_\lambda u=\mu g(z)|u|^{r-1}u+h(z)|u|^{s-1}u \;\text{in}\; \Omega u\in H^{1,\lambda}_0(\Omega) \end{array}\right\} \end{align*} where $\Delta_\lambda=\Delta_x+|x|^{2\lambda}\Delta_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\in\Omega$, $N=n+m;\, n,\, m\geq 1$, $\lambda>0$, $0\leq r<1<s<2^*_\lambda-1$ and $\mu$ is a positive parameter. The functions $g$ and $h$ may change sign and $2^*_\lambda=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q=n+(1+\lambda)m$ of $\Delta_\lambda$. In the critical case $s=2^*_\lambda-1$, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions $g\geq 0$ and $h\equiv 1$.

math.AP↗

arXiv2023

Ground state solutions for quasilinear Schrodinger type equation involving anisotropic p-laplacian

This paper is concerned with the existence of a nonnegative ground state solution of the following quasilinear Schrödinger equation \begin{equation*} \begin{split} -Δ_{H,p}u+V(x)|u|^{p-2}u-Δ_{H,p}(|u|^{2α}) |u|^{2α-2}u=λ|u|^{q-1}u \text{ in }\;R^n;\; u\in W^{1,p}(\;R^n)\cap L^\infty(\;R^N) \end{split} \end{equation*} where $N\geq2$; $(α,p)\in D_N=\{(x,y)\in \;R^2 : 2xy\geq y+1,\; y\geq2x,\; y 0$ is a parameter. The operator $Δ_{H,p}$ is the reversible Finsler p-Laplacian operator with the function $H$ being the Minkowski norm on $\;R^N$. Under certain conditions on $V$, we establish the existence of a non-trivial non-negative bounded ground state solution of the above equation.

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arXiv2023

Semilinear degenerate elliptic equation in the presence of singular nonlinearity

Given $Ω(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $Ω$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear degenerate elliptic equation with a singular nonlinearity given by: \begin{align} -Δ_λu&=\frac{f}{u^ν} \text{ in }Ω\nonumber &u>0 \text{ in } Ω\nonumber &u=0 \text{ on } \partialΩ\nonumber \end{align} where the operator $Δ_λ$ is given by $$Δ_λ{u}=u_{xx}+|x|^{2λ}Δ_y{u};\,(x,y)\in \;R\times\;R^m $$ is known as the Grushin operator.

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