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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains


Author: Song-Sun Lin
Journal: Trans. Amer. Math. Soc. 332 (1992), 775-791
MSC: Primary 35B05; Secondary 35B32, 35J50, 35J65
DOI: https://doi.org/10.1090/S0002-9947-1992-1055571-1
MathSciNet review: 1055571
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Abstract: We study the existence of positive nonradial solutions of equation $ \Delta u + f(u) = 0$ in $ {\Omega _a}$, $ u = 0$ on $ \partial {\Omega _a}$, where $ {\Omega _a} = \{ x \in {\mathbb{R}^n}:a < \vert x\vert < 1\} $ is an annulus in $ {\mathbb{R}^n}$, $ n \geq 2$, and $ f$ is positive and superlinear at both 0 and $ \infty $. We use a bifurcation method to show that there is a nonradial bifurcation with mode $ k$ at $ {a_k} \in (0,1)$ for any positive integer $ k$ if $ f$ is subcritical and for large $ k$ if $ f$ is supercritical. When $ f$ is subcritical, then a Nehari-type variational method can be used to prove that there exists $ {a^{\ast} } \in (0,1)$ such that for any $ a \in ({a^{\ast} },1)$, the equation has a nonradial solution on $ {\Omega _a}$.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1055571-1
Keywords: Nonradial solution, bifurcation method, variational method
Article copyright: © Copyright 1992 American Mathematical Society