On a nonlocal elliptic problem involving critical growth (Q2249862)

This paper deals with a class of nonlocal quasilinear elliptic problems with Dirichlet boundary condition. The main features of this problem are the following: (i) the presence of the nonlinear term \(| u| ^{p^{*}-2}u\), where \(p^{*}\) denotes the critical Sobolev exponent; (ii) the influence of a small perturbation term. The main results in the present paper establish that if the perturbation term is ``small'' then the problem admits at least one weak solution, provided that the positive parameter associated with the critical term is small enough. The proof relies on the fixed point theorem of Carl and Heikkilä applied to a suitable increasing nonlinear operator between Sobolev spaces.