Nodal solutions to critical growth elliptic problems under Steklov boundary conditions (Q1006193)

The paper deals with elliptic problems at critical growth under Steklov boundary conditions of the type \[ \Delta u=|u|^{2^*-2}u \quad\text{in } \Omega,\qquad u_\nu =\delta u \quad \text{on }\partial\Omega, \tag \(*\) \] where \(\Omega\subset \mathbb R^n\) is a bounded domain, \(n\geq3\), \(2^*=2n/(n-2)\) and \(u_\nu\) stands for the outer normal derivative of \(u\) on \(\partial\Omega.\) The parameter \(\delta\in \mathbb R\) is such that when \(\delta=0\) the problem \((*)\) reduces to the Neumann problem, while \((*)\) tends to the Dirichlet problem when \(\delta\to-\infty\), Employing fine estimates on the concentration of Sobolev minimizers on the boundary, the authors prove existence of nontrivial nodal solutions to \((*)\). Moreover, multiplicity result is obtained in the case when \(\Omega\) is a ball, by taking advantage of the explicit form of the Steklov eigenfunctions. These last results are partially extended to the case of fourth order Steklov boundary value problems \[ \Delta^2 u=|u|^{2_*-2}u \quad \text{in } \Omega, \qquad u=0,\quad \Delta u=du_\nu \quad\text{on } \partial\Omega, \] where now \(n\geq 5\), \(2_*=2n/(n-4)\) and \(d\in \mathbb R\).