Executive Summary : | My research primarily centres on the analysis of elliptic partial differential equations (PDEs). During my doctoral studies, I explored rearrangement invariant function spaces, which include Lorentz, Lorentz-Zygmund and Orlicz spaces. The main objective was to determine optimal weight functions for generalized Hardy-Sobolev inequalities and achieve their best constants. In my postdoctoral tenure, recently, I worked on the spectrum of fractional (p,q)-Laplace operators with two parameters and pointwise gradient potential estimates for weak solutions to mixed local-nonlocal operators. Currently, I focus on two different projects in these directions. The first project is related to the regularity theory for inhomogeneous and nonlocal elliptic equations. Regularity theory plays a crucial role in the study of PDEs. For example, it provides fundamental tools to study the existence and uniqueness of solutions for PDEs; it helps to establish the well-posedness of PDEs. In the literature, there is an ample number of works on the regularity theory. Our project is motivated by the De-Giorgi Nash Moser theory, where the Harnack inequality and Holder regularity were established for weak solutions to a general class of elliptic equations. Later, In 1986, Garofalo, Fabes, and Chiarenza established the Harnack inequality and continuity of weak solutions for a linear and local Schrodinger equation with weight in optimal scale invariant function space, namely in Kato class. Due to the presence of the Kato potentials, Holder regularity cannot be expected for the weak solution. In 2001, Biroli introduced a nonlinear version of their approach and proved the Harnack inequality for the p-Laplace operator. All these results are not well known in the nonlocal setup. In recent years, great attention has been given to the study of problems involving fractional operators due to their application in the framework of nonlinear optics, fractional quantum mechanics, etc. In this project, using De-Giorgi's technique, we aim to establish the Harnack inequality and the continuity of weak solutions to certain nonlocal and mixed local-nonlocal Schrodinger equations with appropriate Kato potentials. Our second project is related to the study of the spectrum for fractional (p,q)-Laplace operator with two parameters. Unlike the fractional p-Laplace operator, the spectrum of the fractional (p,q)-Laplace operator is not well studied in the literature. In our recent work, depending on the ranges of the parameters, we provide a complete description of the existence and nonexistence of positive solutions for this eigenvalue problem. The nonexistence of positive solutions opens a question of whether the spectrum of fractional (p,q)-Laplace operator possesses sign-changing solutions. In this project, we try to find the location of the parameters in the two-dimensional plane where the generalized eigenvalue problem possesses sign-changing or nodal solutions. |