Double phase problem with critical and locally defined reaction terms

Abstract

We consider a double phase problem with a reaction consisting of a parametric critical term plus a Caratheodory perturbation which is only locally restricted near zero. We show that for all big values of the parameter $λ > 0$ the problem has at least three nontrivial smooth solutions all with sign information (two of constant sign and the third nodal). Also we establish the asymptotic behavior of the solutions as $λ \to + \infty$ . Finally if we improve the regularity of the perturbation term, we show that there is a second nodal solution for a total of four nontrivial smooth solutions all with sign information.

Keywords

Double phase equation cut-off function constant sign solutions nodal solutions nonlinear regularity critical groups critical term

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