Electron. J. Diff. Equ., Vol. 2010(2010), No. 158, pp. 1-16.

Existence of nonnegative solutions to positone-type problems in R^N with indefinite weights

Dhanya Rajendran, Jagmohan Tyagi

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 4, 2010.
Math Subject Classifications: 35J45, 35J55.
Key Words: Elliptic system; nonnegative solution; existence of solutions.

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  Dhanya Rajendran
TIFR Centre For Applicable Mathematics, Post Bag No. 6503, Sharda Nagar
Chikkabommasandra, Bangalore-560065, Karnataka, India
email: dhanya@math.tifrbng.res.in
Jagmohan Tyagi
TIFR Centre For Applicable Mathematics, Post Bag No. 6503, Sharda Nagar
Chikkabommasandra, Bangalore-560065, Karnataka, India
email: jtyagi1@gmail.com

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