Research Projects

Nonlinear analysis has seen significant development in recent decades, with elliptic partial differential equations (PDEs) playing a critical role in modeling natural phenomena in mathematical physics, optimization, and economy. Mathematicians have always sought to understand the existence and qualitative nature of solutions to these problems. The subject of nonlocal elliptic equations has gained popularity due to their attractive analytical structure and various applications. This project proposes studying nonlocal elliptic equations involving critical growth nonlinearities, which are not always connected and smooth in the real world. It also aims to study solutions in domains with small holes or path connected contractible domains. The study of solutions after adding nonlinear or non-homogeneous perturbations is challenging due to the energy functional associated with the equations not preserving the energy of the solution under the modulus. The second main aim is to study fourth-order elliptic equations with Choquard type nonlinearities, exploring the existence of ground state, regularity, and nonexistence of solutions. The current state of the art on the Choquard equation does not guarantee solutions for these types of equations. The project will use critical point theory, variational methods, algebraic geometry techniques, Polarization, Moving plane methods, and Moving sphere methods. The resulting methods will be beneficial in various fields, such as the modern theory of nonlinear functional analysis, critical point theory, and variational inequalities, and will serve as building blocks for studying other nonlocal partial differential equations.