Nodal solutions for critical Robin double phase problems with variable exponent

Abstract

. In this paper, we study a nonlinear double phase problem with variable exponent and critical growth on the boundary. The problem has in the reaction the combined effects of a Carath´eodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to apply results of the critical point theory to the corresponding energy functional. Thus, we use appropriate cut-off functions and truncation techniques to work on an auxiliary coercive problem. In this way, we can use variational tools to get a sequence of sign changing solutions to our main problem. Further, we show that such a sequence converges to 0 in L ∞ and in the Musielak-Orlicz Sobolev space.