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The existence of a positive solution to asymptotically linear scalar field equations

Published online by Cambridge University Press:  11 July 2007

G. Li  and
H.-S. Zhou
G. Li
Affiliation:
Young Scientist Laboratory of Mathematical Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071, People's Republic of [email protected]
H.-S. Zhou
Affiliation:
Young Scientist Laboratory of Mathematical Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071, People's Republic of [email protected]
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Abstract

We consider the following elliptic equation:where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0,which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 130 , Issue 1 , February 2000 , pp. 81 - 105
Copyright
Copyright © Royal Society of Edinburgh 2000

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