Parametric anisotropic \((p, q)\)-Neumann problems
This paper presents a bifurcation-type theorem describing the changes in the set of positive solutions of the parametric anisotropic Neumann problem \[ \begin{cases} -\Delta_{p(z)} u(z)-\Delta_{q(z)} u(z)+\lambda u(z)^{p(z)-1}=f(z,u(z)) &\text{in }\Omega,\\ \frac{\partial u}{\partial n}=0 & \text{on }\partial\Omega, \end{cases} \] as the parameter \(\lambda\) moves on \(\mathbb{R}_+=(0,+\infty)\), where \(\Omega\subseteq \mathbb{R}^N\) is a bounded domain with \(C^2\)-boundary \(\partial\Omega\), \(p,q\in C^1(\overline{\Omega})\), \(\Delta_{r(z)}\) is the \(r(z)\)-Laplacian when \(r\in C(\overline{\Omega})\) with \(1<r(z)\) for all \(z\in \overline{\Omega}\) and \(f\colon\Omega\times\mathbb{R}\to\mathbb{R}\) is a Carathéodory function such that \(f(z,\cdot)\) is \((p_+-1)\)-superlinear near \(+\infty\) for \(p_+=\max_{\overline{\Omega}}p\) and \(f(z,\cdot)\) is \((q_--1)\)-superlinear near \(0^+\) for \(q_-=\min_{\overline{\Omega}}q\).
scientific article; zbMATH DE number 7767310
scientific article