R. Arora

On the classification of solutions to the Logarithmic Laplacian critical Choquard equation

Rakesh Arora, Jacques Giacomoni

In this work, we establish a sharp logarithmic Choquard inequality by combining Beckner's entropy inequality with the sharp Pitt-type inequality. Motivated by this estimate, we cla…

On fractional $1$ -Laplacian evolution equation

Rakesh Arora, Nitin Kumar Maurya

We study a nonlocal evolution problem driven by the fractional $1$ -Laplacian with a Carathéodory nonlinearity satisfying subcritical growth conditions. We develop a potential well…

Global Calderon-Zygmund estimates for irregular double-phase evolution problem with non-divergence data

Rakesh Arora, Sergey Shmarev

We study irregular double-phase parabolic equations with variable exponents and non-divergence data, \[ u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\q…

Spectral Properties of the Logarithmic Laplacian with Indefinite Weights

Rakesh Arora, Tuhina Mukherjee, Arshi Vaishnavi

In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusterni…

Sharp embeddings and existence results for Logarithmic $p$ -Laplacian equations with critical growth

Rakesh Arora, Jacques Giacomoni, Hichem Hajaiej +1

In this paper, we derive a new $p$ -Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the lo…

Irregular double-phase evolution problem: existence and global regularity

Rakesh Arora, Sergey Shmarev

We investigate the homogeneous Dirichlet problem for the irregular double-phase evolution equation \[ u_t-\operatorname{div} \left( a(z)|\nabla u|^{p(z)-2} \nabla u + b(z)|\nabla u…

Global Existence and Finite-Time Blow-Up of Solutions for Parabolic Equations Involving the Fractional Musielak $g_{x, y}$ -Laplacian

Rakesh Arora, Anouar Bahrouni, Nitin Kumar Maurya

In this work, we study the parabolic fractional Musielak $g_{x, y}$ -Laplacian equation: \begin{equation*} \left\{ \begin{aligned} u_{t} + (-Δ)_{{g}_{x,y}}^{s} u &= f(x,u), && \text{…

The Brezis-Nirenberg and logistic problem for the Logarithmic Laplacian

Rakesh Arora, Jacques Giacomoni, Arshi Vaishnavi

In this work, we study the non-local analogue of Brezis-Nirenberg and logistic type elliptic equations involving the logarithmic Laplacian and critical logarithmic non-linearity wi…

math.AP2024

Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results

Rakesh Arora, Jacques Giacomoni, Arshi Vaishnavi

In this work, we study the existence, non-existence, and uniqueness results for nonlocal elliptic equations involving logarithmic Laplacian, and subcritical, critical, and supercri…

math.AP2024

Global gradient estimates for solutions of parabolic equations with nonstandard growth

Rakesh Arora, Sergey Shmarev

We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equation…

math.AP2023★ 1 cited

Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

Rakesh Arora, Sergey Shmarev

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in…

math.AP2022★ 1 cited

A large class of nonlocal elliptic equations with singular nonlinearities

Rakesh Arora, Phuoc-Tai Nguyen, Vicentiu D. Radulescu

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $L_{γ}^{s} u = F (u)$ , posed in…

math.AP2022

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

Rakesh Arora, Sergey Shmarev

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinde…

On double phase Kirchhoff problems with singular nonlinearity

Rakesh Arora, Alessio Fiscella, Tuhina Mukherjee +1

In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcr…

Combined effects in mixed local-nonlocal stationary problems

Rakesh Arora, Vicentiu D. Radulescu

In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlin…

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

Rakesh Arora, Sergey Shmarev

We study the homogeneous Dirichlet problem for the equation \[ u_t-\operatorname{div}\left((a(z)\vert \nabla u\vert ^{p(z)-2}+b(z)\vert \nabla u\vert ^{q(z)-2})\nabla u\right)=f\qu…