Stability of positive solutions to \(p\)-Laplace type equations (Q2868075)

The author considers the boundary value problem \(- \Delta_p u - \alpha \Delta u = g(x,u)\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\), where \(\alpha\) is a positive real number, \(\Omega \subset\mathbb R^n\) is a bounded domain with a smooth boundary \(\partial \Omega\), \(n \geq 2\), and \(2 \leq p < \infty\). Using the method of lower-upper solution, it is shown that this problem has a solution. Furthermore, it is proved that the problem \(- \Delta_pu - \alpha \Delta u = \lambda(u - f(u))\) in \(\Omega\), \(u=0\) on \(\partial \Omega\) has a positive solution supposed that \(f\) fulfills the following assumptions: \(f \in C(\mathbb R^+,\mathbb R)\) and for any \(t_0 > 0\) there exists a positive number \(A\) such that for all \(t \in [0,t_0]\) the inequality \(| t - f(t) | \leq A\) holds; \(f(0) < 0\) and there exists a positive number \(\beta\) such that \(\beta = f(\beta)\). If additionally \(f^\prime(y) \geq f(y)/y\) for all \(y \in (0,\infty)\), then the positive solution of this problem is stable.