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.