arXiv subjects

Prashanta Garain

Publications and source records attributed to Prashanta Garain.

At least 19 recordsLinked to original sources

arXiv2026

Mixed local and nonlocal weighted singular quasilinear elliptic problem and its associated Sobolev-type inequality

We consider a class of mixed anisotropic and nonlocal singular quasilinear elliptic problem associated with Muckenhoupt weights. The presence of the singular nonlinearity, which blows up near the origin along with its interaction with anisotropy, weighted degeneracy, and nonlocal diffusion creates significant analytical challenges. We employ monotone approximation, weighted Sobolev embeddings, compactness, and variational methods to establish the existence and uniqueness of weak solutions under suitable assumptions on the datum. Further, we characterize the best constant in an associated weighted mixed anisotropic and nonlocal Sobolev-type inequality, prove that it is attained, and show that the normalized weak solution is the unique extremal. These results are new even in the mixed weighted Laplace case $p=2$.

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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

Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities

We study a class of mixed local-nonlocal equations with Hartree-type nonlinearities of the form \begin{equation}\label{meqnab} -\Delta u + (-\Delta)^s u + u = (I_\alpha * F(u))\,F'(u) \quad \text{in } \mathbb{R}^N, \end{equation} where $N \geq 3$, $s \in (0,1)$, and $F \in C^1(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions type assumptions. The equation combines the classical Laplacian with the fractional Laplacian, while the Hartree-type nonlinearity is given by a nonlocal convolution term involving the Riesz potential $I_\alpha$, with $\alpha \in (0,N)$. We prove the existence of ground state solutions. To this end, we establish regularity properties and derive a Poho\v{z}aev-type identity for general weak solutions. Moreover, we obtain symmetry properties of ground state solutions via polarization methods.

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arXiv2025

Gradient higher integrability of bounded solutions to parabolic double-phase systems

We prove that bounded solutions to degenerate parabolic double-phase problem modelled upon \[u_t-\dv(|\na u|^{p-2}\na u+a(x,t)|\na u|^{q-2}\na u)=-\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\,, \] where a nonnegative weight $a$ is $\alpha$-H\"older continuous in space and $\tfrac \alpha 2$-H\"older continuous in time, have locally higher integrable gradients for the sharp range of exponents $p<q\le p+\alpha$.

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arXiv2025

Mixed local-nonlocal $p$-Laplace equation with variable singular nonlinearity in the Heisenberg group

We investigate a mixed local-nonlocal $p$-Laplace equation on the Heisenberg group, where the nonlinear term features a variable singular exponent. Our analysis establishes the existence, uniqueness, and regularity of weak solutions under suitable structural assumptions. To the best of our knowledge, this work provides the first treatment of such mixed local-nonlocal problems in a non-commutative setting, even in the linear case $p=2$ with a constant singular exponent.

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arXiv2025

On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group

In this work, we establish a mixed local--nonlocal Sobolev-type inequality in the Heisenberg group and demonstrate that its extremals coincide with solutions to the corresponding mixed local--nonlocal singular $p$-Laplace equations. We further show that these inequalities serve as a necessary and sufficient condition for the existence of weak solutions to the associated singular problems. Notably, the same characterization remains valid in both the purely local and purely nonlocal settings. Our results thus provide a unified framework linking the existence theory for singular equations across local, nonlocal, and mixed regimes.

math.AP↗

arXiv2025

Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity

We study a class of mixed local and nonlocal $p$-Laplace equations with prototype \[ -\Delta_p u + (-\Delta_p)^s u = f \quad \text{in } \Omega, \] where $\Omega \subset \mathbb{R}^n$ is bounded and open. We provide sufficient condition on $f$ to ensure weak Harnack inequality with a tail term for sign-changing supersolutions. Two different proofs are presented, avoiding the Krylov--Safonov covering lemma and expansion of positivity: one via the John--Nirenberg lemma, the other via the Bombieri--Giusti lemma. To our knowledge, these approaches are new, even for $p = 2$ with $f \equiv 0$, and include a new proof of the reverse H\"older inequality for supersolutions. Further, we establish Harnack inequality for solutions by first deriving a local boundedness result, together with a tail estimate and an initial weak Harnack inequality.

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arXiv2025

Higher H\"older regularity for fractional $(p,q)$-Laplace equations

We study the fractional $(p,q)$-Laplace equation $$ (-\Delta_p)^s u +(-\Delta_q)^t u= 0 $$ for $s,t\in(0,1)$ and $p,q\in(1,\infty)$. We establish H\"older estimates with an explicit exponent. As a consequence, we derive a Liouville-type theorem. Our approach builds on techniques previously developed for the fractional $p$-Laplace equation, relying on a Moser-type iteration for difference quotients.

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arXiv2025

Nonlocal singular problem and associated Sobolev type inequality with extremal in the Heisenberg group

We study a fractional $p$-Laplace equation involving a variable exponent singular nonlinearity in the framework of the Heisenberg group. We first establish the existence and regularity of weak solutions. In the case of a constant singular exponent, we further prove the uniqueness of solutions and characterize the extremals of a related Sobolev-type inequality. Additionally, we demonstrate a connection between the solutions of the singular problem and these extremals. To the best of our knowledge, these findings provide new insights even in the classical case $p=2$.

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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.

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arXiv2025

Pohozaev-type identities for classes of quasilinear elliptic local and nonlocal equations and systems, with applications

In this article, we establish Pohozaev-type identities for a class of quasilinear elliptic equations and systems involving both local and nonlocal $p$-Laplace operators. Specifically, we obtain these identities in $\mathbb{R}^n$ for the purely anisotropic $p$-Laplace equations, the purely fractional $p$-Laplace equations, as well as for equations that incorporate both anisotropic and fractional $p$-Laplace features. We also extend these results to the corresponding systems. To the best of our knowledge, the identities we derive in the mixed case are new even when $p=2$. Finally, we illustrate some of the applications of our main results.

math.AP↗

arXiv2025

Estimates of variational eigenvalues on metric measure spaces

In the article, we study variational eigenvalues on doubling metric measure spaces. We prove existence of minimizers of variational Neumann $(p,q)$-eigenvalues on metric measure spaces and on this base we obtain estimates of Neumann eigenvalues.

math.AP↗

arXiv2024

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

Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization

In this article, we present the symmetry of weak solutions to a mixed local-nonlocal singular problem. We also establish results related to the existence, nonexistence, and regularity of weak solutions to a mixed local-nonlocal singular jumping problem. A crucial element in proving our main results is the variational characterization of the solutions, which also reveals the decomposition property. This decomposition property, together with comparison principles and the moving plane method, yields the symmetry result. Additionally, we utilize nonsmooth critical point theory alongside the variational characterization to analyze the jumping problem.

math.AP↗

arXiv2024

Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations

This article is divided into two parts. In the first part, we examine the Brezis-Oswald problem involving a mixed anisotropic and nonlocal $p$-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal $p$-Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal $p$-Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest.

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arXiv2024

On the Neumann $(p,q)$-eigenvalue problem in Hölder singular domains

In the article we study the Neumann $(p,q)$-eigenvalue problems in bounded Hölder $γ$-singular domains $Ω_γ\subset \mathbb{R}^n$. In the case $1<p<\infty$ and $1<q<p^{*}_γ$ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of $(p,q)$-eigenvalues.

math.AP↗

arXiv2024

Higher Hölder regularity for the fractional $p$-Laplace equation in the subquadratic case

We study the fractional $p$-Laplace equation $$ (-Δ_p)^s u = 0 $$ for $0<s<1$ and in the subquadratic case $1<p<2$. We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$. The arguments are based on a careful Moser-type iteration and a perturbation argument.

math.AP↗

arXiv2024

Higher Hölder regularity for a subquadratic nonlocal parabolic equation

In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation $$ \partial_t u+(-Δ_p)^s u = 0. $$ Here, $(-Δ_p)^s$ is the fractional $p$-Laplacian, $0<s<1$ and $1<p<2$. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when $p\geq 2$.

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