Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms
Given a bounded domain \(\Omega \subseteq \mathbb{R}^N\) with a \(C^2\)-boundary \(\partial\Omega\), the authors study the following semilinear Robin problem \[ \begin{alignedat}{2} -\Delta u(z) +\xi(z) u(z) & = \vartheta(z) |u(z)|^{q-2}u(z)+f(z,u(z))\quad \text{in}\quad \Omega,\\ \frac{\partial u}{\partial n}+\beta(z)u & = 0\quad \text{on}\quad \partial \Omega, \end{alignedat}\tag{1} \] where \(1<q<2\), \(\xi \in L^s(\Omega)\) with \(s>N\) being sign-changing, \(\vartheta \in L^\infty(\Omega)\), \(\vartheta(z)>0\) for almost all \(z\in\Omega\) and \(f:\Omega\times\mathbb{R}\to\mathbb{R}\) is a Carathéodory function of arbitrary growth. By applying a variant of the symmetric mountain pass theorem, the authors show the existence of smooth nodal (i.e., sign-changing) solutions which converge to zero in \(C^1(\overline{\Omega})\). Furthermore, if the coefficient of the concave term (that is, \(\vartheta(z) |u(z)|^{q-2}u(z)\), \(1<q<2\)) is sign changing, then it is shown again the existence of a sequence of smooth solutions converging to zero in \(C^1(\overline{\Omega})\) but without any knowledge about the sign.
scientific article; zbMATH DE number 6851000