Nonnegative solutions for a class of radially symmetric nonpositone problems

Abstract:

We consider the existence of radially symmetric non-negative solutions for the boundary value problem \[ \begin {array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\quad \left \| x \right \| \leq 1,x \in {R^N}(N \geq 2)} \\ {u(x) = 0\quad \left \| x \right \| = 1} \\ \end {array} \] where $\lambda > 0,f(0) < 0$ (non-positone), $f’ \geq 0$ and $f$ is superlinear. We establish existence of non-negative solutions for $\lambda$ small which extends some work of our previous paper on non-positone problems, where we considered the case $N = 1$. Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.