Geodesic
Existence of nonnegative solutions to positone-type problems in \(\mathbb R^N\) with indefinite weights
Electronic journal of differential equations, Tome 2010 (2010)
We study the existence of a nonnegative solution to the following problem in

$\displaylines{ -\Delta u=\lambda a(x)f(u) \cr u(x) \to 0 \quad \hbox{as }|x|\to \infty. }$

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.
Classification : 35J45, 35J55
Keywords: elliptic system, nonnegative solution, existence of solutions 
@article{EJDE_2010__2010__a98,
     author = {Rajendran,  Dhanya and Tyagi,  Jagmohan},
     title = {Existence of nonnegative solutions to positone-type problems in \(\mathbb {R^N\)} with indefinite weights},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1205.35102},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a98/}
}
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%J Electronic journal of differential equations
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%F EJDE_2010__2010__a98
Rajendran,  Dhanya; Tyagi,  Jagmohan. Existence of nonnegative solutions to positone-type problems in \(\mathbb R^N\) with indefinite weights. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a98/