Positive solutions for singular superlinear \(\phi\)-Laplacian problems with nonlinear boundary conditions
The paper deals with positive solutions of the boundary value problem \[ \begin{cases} -(\phi(u'))' = \lambda h(t) f(u), \\ u(0)=0, \quad u'(1)+H(u(1))=0, \end{cases} \] where \(\lambda>0\) is a parameter, \(\phi\colon\mathbb{R}\to\mathbb{R}\) is an odd increasing homeomorphism, \(h \colon (0,1]\to (0,+\infty)\) is continuous, \(f \colon (0,+\infty)\to (0,+\infty)\) is continuously differentiable with \(\inf f>0\), and \(\phi\)-superlinear at \(+\infty\), with possible singularity at \(0\), \(H\colon [0,+\infty)\to [0,+\infty)\) is continuous, nondecreasing, with \(H(0)=0\). Using the fixed point theory and the topological degree, the authors prove the existence of \(\lambda^{*}>0\) such that there are at least two (ordered) solutions for \(\lambda<\lambda^{*}\), at least one solution for \(\lambda=\lambda^{*}\), and no solution for \(\lambda>\lambda^{*}\).
scientific article; zbMATH DE number 7465533