Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential.
The authors study the following nonlinear elliptic problem \[ \begin{gathered} -\text{div}(\| Dx(z)\|^{p- 2}Dx(z))- \lambda,\;| x(z)|^{p- 2}x(z)\in \partial_j(z, x(z))\text{ a.e. on }Z,\\ x|_\Gamma= 0,\end{gathered}\tag{1} \] where \(Z\in\mathbb{R}^N\) is a bounded domain with a \(C^{1,\alpha}\)-boundary \(\Gamma\) \((\alpha\in (0,1))\) and \(j(z,\cdot)\) is a locally Lipschitz not necessarily smooth potential, with \(\partial j(z,\cdot)\) being the subdifferential in the sense of Clarke. The goal of the authors is to establish the existence of at least two distinct solutions for (1) assuming a different asymptotic behaviour of \(\partial j(z,\cdot)\) at \(\pm\infty\). Moreover, the authors show that one of the solutions is smooth and strictly positive.
scientific article; zbMATH DE number 2062878
scientific article