Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity, stability and asymptotic profiles (Q502257)

The authors study the existence, multiplicity and asymptotic behavior of nontrivial solutions for problems \[ -\Delta u = \lambda \left(m(x)u - a(x)|u|^{p-2} u\right) \;\;\text{ in } \;\;\Omega, \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(N\geq 2\), with boundary conditions \[ \frac{\partial u}{\partial {\mathbf n}}= \lambda b(x)|u|^{q-2}u \;\;\text{ on } \;\;\partial \Omega. \]