Electronic Journal of Differential Equations, Vol. 1994(1994), No. 04, pp. 1-10. Title: Existence Results for Non-autonomous Elliptic Boundary Value Problems Authors: V. Anuradha (Univ. of Arkansas at Little Rock, USA) S. Dickens (Mississippi State Univ., MS, USA) R. Shivaji (Mississippi State Univ., MS, USA) Abstract: We study solutions to the boundary value problems $$-\Delta u(x) = \lambda f(x, u);\quad x \in \Omega$$ $$u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega$$ where $\lambda > 0$, $\Omega$ is a bounded region in $\Bbb{R}^N$; $N \geq 1$ with smooth boundary $\partial \Omega$, $\alpha(x) \geq 0$, $n$ is the outward unit normal, and $f$ is a smooth function such that it has either sublinear or restricted linear growth in $u$ at infinity, uniformly in $x$. We also consider $f$ such that $f(x, u) u \leq 0$ uniformly in $x$, when $|u|$ is large. Without requiring any sign condition on $f(x, 0)$, thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given $\lambda \in (\lambda_{n}, \lambda_{n + 1})$ where $\lambda_{k}$ is the $k$-th eigenvalue of $-\Delta$ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for $\lambda$ near $\lambda_{1}$, and for $\lambda$ large when $f$ is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for $\lambda$ small. Submitted January 23, 1994. Published July 8, 1994. Math Subject Classification: 35J65. Key Words: Elliptic boundary value problems; semipositone.