A Study on Variational Problems Related to Sobolev Spaces

Abstract

In this thesis, we prove two generalized concentration-compactness principles for variable exponent Lebesgue spaces and as an application study the following prob lem: U∗ ϵ(p(.), q(.), Λ) := sup U(m) Λ 0 ϵq(s) ds : m ∈ W1,p(.) (Λ),∥∇m∥Lp(.)(Λ) ≤ ϵ , (1) Where U : R → R is an upper semicontinuous, non zero in the L1 sense, 0 ≤ U(m) ≤ c|m|q(.), Λ is a bounded subset of Rn, n ≥ 3, 1 < p(.) < n, and p(.) ≤ q(.) ≤ p∗(.). Moreover, we also study this problem with variational techniques like Gamma convergence