Monotonicity of the forcing term and existence of positive solutions for a class of semilinear elliptic problems

Abstract:

We study the existence of positive solutions to the equation $\Delta u + f(u) + \lambda g(\left \| x \right \|) = 0$ in the unit ball in ${\mathbb {R}^N}$ with Dirichlet boundary conditions, where $f$ is superlinear with $f(0) = 0$ and $\lambda$ is a real parameter. We prove that if $g$ is monotonically increasing, then there exists an $\alpha < 0$ such that for $\lambda < \alpha$ the above equation has no positive solution. This is in contrast to the case of $g$ monotonically decreasing, where positive solutions exist for all negative values of $\lambda$.