Singular limits solution for 2-dimensional elliptic problems involving exponential nonlinearities with sub-quadratic convection nonlinear gradient terms and singular weights

摘要

Given abounded open regular set ? of R~2, q_1...,q_K∈ ?, a regular bounded function q:?→ [0, +∞) and a bounded function V: Ω → [0, +∞), we give a sufficient condition for the model problem -?w - λe(x)|su|~q = ε~2V(x)e~u to have a positive weak solution in Ω with u = 0 on ?Ω, which is singular at each q_i as the parameters ε and λ tend to 0, without considering any relation between them, essentially when the set of concentration points q_i and the set of zeros of V are not necessarily disjoint and q ∈ [1,2) is a real number.