### B Ko and K J Brown

#### Abstract

We discuss the existence of positive classical solutions of the boundary value problem: \begin{center} $-\Delta u=\lambda g(x)f(u) \mbox{ in }\Omega; \hspace{.2in} \displaystyle (1-\alpha)\frac{\partial u}{\partial n}+\alpha u=0 \mbox{ on } \partial\Omega$ \end{center} where $\lambda$ and $\alpha < 1$ are real parameters,$\Omega$ is an open bounded region of $R}^N, \, N \geq 2$ with smooth boundary and $g:\overline \Omega \to \bf R$ is a smooth function which changes sign on $\Omega$ in the cases where $f(u) = u(1 - |u|^p)$ and $f(u) = u(1 + |u|^p)$. \par A constrained variational principle is used to prove the existence of positive solutions over a certain range of $\lambda$ which may contain both positive and negative values of $\lambda$; this variational principle depends on the spectral properties of the corresponding linearized problem. \par The solutions obtained in the case $f(u) = u(1 - |u|^p)$ do not lie on a branch of solutions bifurcating from the trivial branch of zero solutions at a principal eigenvalue; such solutions do not exist in the case where $g$ does not change sign and so the results show the crucial role played by the presence of the indefinite weight function in the existence of positive solutions.