On a class of critical Robin problems

Abstract

Abstract We consider a nonlinear parametric Robin problem. In the reaction, there are two terms, one critical and the other locally defined. Using cut-off techniques, together with variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least three nontrivial smooth solutions all with sign information, which converge to zero in C 1 ⁢ ( Ω ¯ ) {C^{1}(\bar{\Omega})} as the parameter λ → 0 + {\lambda\to 0^{+}} .