Physical Sciences and Mathematics Commons^™

Nonnegative Solutions For A Class Of Radially Symmetric Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the existence of radially symmetric non-negative solutions for the boundary value problem

\begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\qua... ...\\ {u(x) = 0\quad \left\Vert x \right\Vert = 1} \\ \end{array} \end{displaymath}

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.

Nonnegative Solutions To A Semilinear Dirichlet Problem In A Ball Are Positive And Radially Symmetric, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We prove that nonnegative solutions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular, this answers a question in [3] where positive solutions were proven to be radially symmetric. In section 4 we provide a sufficient condition on the geometry of the domain which ensures that nonnegative solutions are positive in the interior.