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

• Main Authors: Petru Jebelean, Calin-Constantin Serban
• Format: Article
• Language: English
• Published: University of Szeged 2020-12-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: minkowski operator; fisher–kolmogorov nonlinearities; krasnoselskii's genus; critical point
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=8943

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.