Research Projects

This project aims to investigate the existence, multiplicity, and concentration profile of solutions in strongly coupled elliptic systems, focusing on equations from physical and mathematical models like switched diffusion process and stochastic control. The project will investigate sign-changing solutions for semi-linear and quasilinear elliptic systems and systems with critical nonlinearity, which is delicate to handle. The study will also examine the generalized spectrum of the corresponding eigenvalue problem, which plays a crucial role in the existence of single and multiple solutions. The second part of the project will study a perturbed system and the solutions' behavior as the perturbation parameter converges to zero. The convergence of the corresponding sequence of solutions is a natural question in approximation theory, and the concentration profile depends on the boundary data described for the problem. Studies for positive solutions have been conducted recently, but little is known about the concentration profile of sign-changing solutions. The goal is to study the existence and concentration profile of sign-changing solutions to better understand the theory.