Existence and nonexistence of positive solutions for a class of superlinear semipositone systems
We consider an elliptic system of the form \[ \begin{aligned} -\Delta u=\lambda f(v) &\quad\text{in }\Omega,\\ -\Delta v=\lambda g(u) &\quad\text{in }\Omega,\\ u=0=v &\quad\text{on }\partial\Omega, \end{aligned} \] where \(\lambda>0\) is a parameter, \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\). Here the nonlinearities \(f,g:[0,\infty)\to\mathbb R\) are \(C_{\text{loc}}^{0,\sigma}\), \(0<\sigma<1\), functions that are superlinear at infinity and satisfy \(f(0)<0\) and \(g(0)<0\). We prove that the system has a positive solution for \(\lambda\) small when \(\Omega\) is convex with \(C^3\) boundary and no positive solution for \(\lambda\) large when \(\Omega\) is a general bounded domain with \(C^{2,\beta}\) boundary. Moreover, we show that there exists a closed connected subset of positive solutions bifurcating from infinity at \(\lambda=0\).
scientific article; zbMATH DE number 5597893
scientific article