Positive solutions for parametric \((p(z),q(z))\)-equations
Let \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega\). The authors study a parametric Dirichlet \((p,q)\)-Laplacian problem of the form \[ \begin{cases} -\Delta_{p(z)}u(z)- \Delta_{q(z)}u(z)= \lambda f(z,u(z))\quad \mbox{in } \Omega,\\ u|_{\partial \Omega}=0, \quad u> 0 ,\quad \lambda>0, \end{cases}\tag{\(P_\lambda\)}\] where \(p,q \in C^{0,1}(\overline{\Omega})\) (that is, \(p,q\) are Lipschitz continuous) satisfying the condition \(1<q_-\leq q_+<p_-\leq p_+\) (recall the notation \(r_-=\min_{\overline{\Omega}}r\) and \(r_+=\max_{\overline{\Omega}}r\)), and \(\Delta_r(z) u= \operatorname{div}(|\nabla u|^{r(z)-2}\nabla u)\) for all \(u \in W_0^{1,r(r)}(\Omega)\) is the \(r(z)\)-Laplace operator. Moreover, \(f:\Omega \times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function satisfying useful properties. In particular \(f\) is \((p_+-1)\)-superlinear in the \(x\)-variable, but without satisfying the usual Ambrosetti-Rabinowitz condition. Thus, the authors develop a variational approach to study positive solutions of \((P_\lambda)\). They prove a bifurcation-type theorem describing the dependence of the set of positive solutions as the parameter \(\lambda>0\) varies.
scientific article; zbMATH DE number 7435416
scientific article