Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 158, pp. 1-16.
Title: Existence of nonnegative solutions to positone-type problems
in R^N with indefinite weights
Authors: Dhanya Rajendran (TIFR, Karnataka, India)
Jagmohan Tyagi (TIFR, Karnataka, India)
Abstract:
We study the existence of a nonnegative solution to the following
problem in
${\mathbb{R}^N}$, $N \geq 3$, in both the radial as well as in the
non-radial case with an indefinite weight function $a(x)$:
$$\displaylines{
-\Delta u=\lambda a(x)f(u) \cr
u(x) \to 0 \quad \hbox{as }|x|\to \infty.
}$$
The nonlinearity f above is of "positone" type; i.e., f is
monotone increasing with $f(0)>0$. We show the existence of a
nonnegative solution to the above problem for $\lambda>0$ small enough.
We also prove the existence of a nonnegative solution to the above
problem in exterior as well as in annular domains. Motivated by the
scalar equation, we further extend these results to the case of
coupled system. Our proof involves the method of monotone iteration
applied to the integral equation corresponding to the problem.
Submitted January 16, 2010. Published November 04, 2010.
Math Subject Classifications: 35J45, 35J55.
Key Words: Elliptic system; nonnegative solution; existence of solutions.