Nodal solutions to critical growth elliptic problems under Steklov boundary conditions

Abstract

We study elliptic problems at critical growth under Steklovboundary conditions in bounded domains. For a second order problem we proveexistence of nontrivial nodal solutions. These are obtained bycombining a suitable linking argument with fine estimates on theconcentration of Sobolev minimizers on the boundary. When the domain is theunit ball, we obtain a multiplicity result by taking advantage of the explicit formof the Steklov eigenfunctions. We also partially extend the results in theball to the case of fourth order Steklov boundary value problems.