arXiv subjects

Deepak Kumar Mahanta

Publications and source records attributed to Deepak Kumar Mahanta.

5 recordsLinked to original sources

arXiv2024

A Note on Sharpened Singular Adams-Type Inequalities

We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.

math.AP↗

arXiv2024

On singularly perturbed $(p, N )$-Laplace Schr\"{o}dinger equation with logarithmic nonlinearity

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a $(p, N)$-Laplace Schr\"{o}dinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space $\mathbb{R}^N$. Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik-Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of the set in which the potential function achieves its minimum values.

math.AP↗

arXiv2024

On the study of $(p, Q)$-Laplace Choquard equations with critical Trudinger-Moser nonlinearity in $\mathbb{H}^N$

This paper deals with the existence and multiplicity of nontrivial solutions for $(p, Q)$-Laplace equations with the Stein-Weiss reaction under critical exponential nonlinearity in the Heisenberg group $\mathbb{H}^N$. In addition, a weight function and two positive parameters have also been included in the nonlinearity. The developed analysis is significantly influenced by these two parameters. Further, the mountain pass theorem, the Ekeland variational principle, the Trudinger-Moser inequality, the doubly weighted Hardy-Littlewood-Sobolev inequality and a completely new Brézis-Lieb type lemma for Choquard nonlinearity play key roles in our proofs.

math.AP↗

arXiv2023

Degenerate Schr{ö}dinger-Kirchhoff $(p, N)$-Laplacian problem With singular Trudinger-Moser nonlinearity in $\mathbb{R}^N$

In this paper, we deal with the existence of nontrivial nonnegative solutions for a $(p, N)$-Laplacian Schr{ö}dinger-Kirchhoff problem in $\mathbb{R}^N$ with singular exponential nonlinearity. The main features of the paper are the $(p, N)$ growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger-Moser inequality, and a completely new Brézis-Lieb type lemma for singular exponential nonlinearity.

math.AP↗

arXiv2023

On the extreme value of Nehari manifold for nonlocal singular Schr{ö}dinger-Kirchhoff equations in $\mathbb{R}^N$

This article investigates the existence, non-existence, and multiplicity of weak solutions for a parameter-dependent nonlocal Schrödinger-Kirchhoff type problem on $\mathbb R^N$ involving singular non-linearity. By performing fine analysis based on Nehari submanifolds and fibre maps, our goal is to show the problem has at least two positive solutions even if $λ$ lies beyond the extremal parameter $λ_\ast$.

math.AP↗