On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

Abstract:

This paper is concerned with the multiplicity of radially symmetric solutions $u(x)$ to the Dirichlet problem \[ \Delta u+f(u)=h(x)+c\phi (x)\] on the unit ball $\Omega \subset \mathbf R^N$ with boundary condition $u=0$ on $\partial \Omega$. Here $\phi (x)$ is a positive function and $f(u)$ is a function that is superlinear (but of subcritical growth) for large positive $u$, while for large negative $u$ we have that $f’(u)<\mu$, where $\mu$ is the smallest positive eigenvalue for $\Delta \psi +\mu \psi =0$ in $\Omega$ with $\psi =0$ on $\partial \Omega$. It is shown that, given any integer $k\ge 0$, the value $c$ may be chosen so large that there are $2k+1$ solutions with $k$ or less interior nodes. Existence of positive solutions is excluded for large enough values of $c$.