Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in the Lorentz-Minkowski space
DOI:
https://doi.org/10.12775/TMNA.2025.030Keywords
Mean curvature operator, radial solutions, concave-convex nonlinearity, variational methods, non-smooth functionalsAbstract
We prove the existence of multiple radial solutions for a class of nonlinear equations - involving the mean curvature operator in the Lorentz-Minkowski space - of the form \begin{equation*} -\dive \bigg(\frac{\nabla u}{\sqrt{1 - \abs{\nabla u}^{2}}}\bigg) = \lambda b(\abs{x})\abs{u}^{q-2}u + f(\abs{x}, u) \quad \text{in } B_{R}, \end{equation*} with Dirichlet boundary conditions in case $q \in (1,2)$ and $f(\abs{x}, s)$ is superlinear in $s$. Solutions are found using Szulkin's critical point theory for non-smooth functionals. Multiplicity results are also given for some cases in which $f$ depends also on (the norm of) the gradient of $u$.References
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