Fisher–Kolmogorov type perturbations of the mean curvature operator in Minkowski space

Abstract

We provide a complete description of the existence/non-existence and multiplicity of distinct pairs of nontrivial solutions to the problem with Minkowski operator $$ -\mbox{div} \left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \lambda u(1-a |u|^q) \quad \mbox{ in } \Omega, \; \; u|_{\partial \Omega}=0, \quad (a\geq0<q),$$ when $\lambda \in (0,\infty)$, in terms of the spectrum of the classical Laplacian. Beforehand, we obtain multiplicity of solutions for parameterized and non-parameterized Dirichlet problems involving odd perturbations of this operator. The approach relies on critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals.

Keywords

• minkowski operator
• fisher–kolmogorov nonlinearities
• krasnoselskii's genus
• critical point