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sub- and supersolutions method

Proceedings of the Edinburgh Mathematical Society (1)

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EXISTENCE OF SOLUTIONS FOR A SYSTEM INVOLVING SCHRÖDINGER OPERATORS WITH WEIGHTS
Laure Cardoulis
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 50 / Issue 3 / October 2007

Published online by Cambridge University Press:

08 January 2008, pp. 611-635
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In this paper, we obtain some results on the existence of solutions for the system
$$ (-\Delta+q_i)u_i=\mu_im_iu_i+f_i(x,u_1,\dots,u_n)\text{ in }\mathbb{R}^{N},\quad i=1,\dots,n, $$
where each of the $q_i$ are positive potentials satisfying $\lim_{|x|\rightarrow+\infty}q_i(x)=+\infty$, each of the $m_i$ are bounded positive weights and each of the $\mu_i$ are real parameters. Depending upon the hypotheses on $f_i$, we use either the method of sub- and supersolutions or a bifurcation method.