P. Garain

Mixed local-nonlocal eigenvalue problems in the Heisenberg group: spectral theory and singular multiplicity

Prashanta Garain, Vicentiu Radulescu

This paper investigates nonlinear eigenvalue problems governed by a family of operators that combine two distinct diffusion mechanisms: a classical local diffusion, which accounts…

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

Prashanta Garain

We consider a class of mixed anisotropic and nonlocal singular quasilinear elliptic problem associated with Muckenhoupt weights. The presence of the singular nonlinearity, which bl…

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

Sanjit Biswas, Prashanta Garain

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, yi…

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

Gurdev Chand Anthal, Prashanta Garain, Nidhi Nidhi

We study a class of mixed local-nonlocal equations with Hartree-type nonlinearities of the form \begin{equation}\label{meqnab} -Δu + (-Δ)^s u + u = (I_α* F(u))\,F'(u) \quad \text{i…

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

Prashanta Garain

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 m…

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

Prashanta Garain

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 exi…

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

Iwona Chlebicka, Prashanta Garain, Wontae Kim

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)\…

Higher Hölder regularity for fractional $(p, q)$ -Laplace equations

Prashanta Garain, Erik Lindgren

We study the fractional $(p, q)$ -Laplace equation $(- Δ_{p})^{s} u + (- Δ_{q})^{t} u = 0$ for $s, t \in (0, 1)$ and $p, q \in (1, \infty)$ . We establish Hölder estimates with an explicit exponent.…

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

Prashanta Garain

We study a class of mixed local and nonlocal $p$ -Laplace equations with prototype \[ -Δ_p u + (-Δ_p)^s u = f \quad \text{in } Ω, \] where $Ω \subset R^{n}$ is bounded and ope…

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

Prashanta Garain

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 regula…

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

Sanjit Biswas, Prashanta Garain

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 treatmen…

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

Gurdev Chand Anthal, Prashanta Garain

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. Specifical…

Estimates of variational eigenvalues on metric measure spaces

Prashanta Garain, Alexander Ukhlov

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 sp…

math.AP2024★ 2 cited

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

Sanjit Biswas, Prashanta Garain

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 pr…

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

Gurdev Chand Anthal, Prashanta Garain

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 regular…

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

Prashanta Garain

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 result…

Higher Hölder regularity for a subquadratic nonlocal parabolic equation

Prashanta Garain, Erik Lindgren, Alireza Tavakoli

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…

On the weighted Steklov eigenvalue problems in outward cuspidal domains

Prashanta Garain, Vladimir Gol'dshtein, Alexander Ukhlov

In this article, we investigate the weighted Steklov eigenvalue problem and the weighted Schrödinger--Steklov eigenvalue problem in outward cuspidal domains. We prove the solvabili…

On $(p, q)$ -eigenvalues of the weighted $p$ -Laplace operator in outward Hölder cuspidal domains

Prashanta Garain, Valerii Pchelintsev, Alexander Ukhlov

In this paper, we will study Neumann $(p, q)$ -eigenvalue problem for the weighted $p$ -Laplace operator in outward Hölder cuspidal domains. The suggested method is based on the compo…

math.AP2023★ 1 cited

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

Prashanta Garain, Erik Lindgren

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. Th…

On the regularity theory for mixed anisotropic and nonlocal $p$ -Laplace equations and its applications to singular problems

Prashanta Garain, Wontae Kim, Juha Kinnunen

We establish existence results for a class of mixed anisotropic and nonlocal $p$ -Laplace equation with singular nonlinearities. We consider both constant and variable singular expo…

Mixed local and nonlocal Dirichlet $(p, q)$ -eigenvalue problem

Prashanta Garain, Alexander Ukhlov

In this article, we consider the spectral problem for the mixed local and nonlocal $p$ -Laplace operator. We discuss the existence and regularity of eigenfunction of the associated…

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

Prashanta Garain, Valerii Pchelintsev, Alexander Ukhlov

In the article we study the Neumann $(p, q)$ -eigenvalue problems in bounded Hölder $γ$ -singular domains $Ω_{γ} \subset R^{n}$ . In the case $1 < p < \infty$ and $1 < q < p_{γ}^{*}$ we prov…