Executive Summary : | This proposal aims to study double phase problems with nonlinearity, concave-convex growth, or non-isolated singularity. The name "double phase problem" comes from the existing two phases governed by a term μ(x), which can be fused with a Kirchhof term for nonlocal variation. These problems have applications in duality theory and the Lavrentiev gap phenomenon. Another class of problem is involving fractional Laplacian operators, which occur in non-local moving fronts, fractal Burgers equations, non-linear stochastic differential equations, mean field and kinetic equations, reaction-diffusion equations, and quasi geostrophic flow. The project can be divided into two parts: problems with regular nonlinearity and singular nonlinearity of the type 1/u^q, q ∈(0, 1). The main obstacle in dealing with singular problems is the non-differentiability of the variational functional. The proposal aims to explore the possibility of multiple positive solutions even when nonlinearity fails to have classical concave-convex behavior, based on the growth assumption on the Kirchhoff term. The novelty of the proposal is to study the multiplicity of solutions beyond the extremal parametric value involved in the problem, which has been little explored in this direction. |