Electron. J. Diff. Eqns.,
Vol. 1994(1994), No. 04, pp. 110.
Existence results for nonautonomous elliptic boundary value problems
V. Anuradha, S. Dickens, & R. Shivaji
Abstract:
We study solutions to the boundary value problems
where
,
is a bounded region in
;
with smooth boundary
,
, 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
uniformly in
x, when u is large. Without requiring any sign condition
on
, thus allowing for both positone as well as
semipositone structure, we discuss the existence of at least
three solutions for given
where
is the kth eigenvalue of
subject
to the above boundary conditions.
In particular, one of the solutions we obtain has nonzero
positive part, while another has nonzero negative part.
We also discuss the existence of three solutions where one of
them is positive, while another is negative, for
near
, and for
large when f is sublinear. We use the method of subsuper
solutions to establish our existence results. We further
discuss nonexistence results for
small.
Submitted January 23, 1994. Published July 8, 1994.
Math Subject Classification: 35J65.
Key Words: Elliptic boundary value problems, semipositone.
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V. Anuradha
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, AR 722041099, USA 

S. Dickens
Department of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762, USA 

R. Shivaji
Department of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762, USA
email: shivaji@math.msstate.edu 
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