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Multiplicity of asymmetric solutions for nonlinear elliptic problems

Author(s): Daomin Cao; Ezzat S. Noussair; Shusen Yan
Journal: Quart. Appl. Math. 64 (2006), 463-482.
MSC (2000): Primary 35J65, 35B25; Secondary 35Q60
Posted: June 21, 2006
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Abstract: In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem:

\begin{displaymath} \begin{cases} -\Delta u+(\lambda-h(x))u=(1-f(x))u^p, &\quad ... ...x\in\mathbb{R}^N, \qquad u\in H^1(\mathbb{R}^N), \end{cases}\end{displaymath}

where $ \lambda>0$ is a parameter, $ h(x)$ and $ f(x)$ are nonnegative radially symmetric functions in $ L^\infty(\mathbb{R}^N)$, $ h(x)$ and $ f(x)$ have compact support in $ \mathbb{R}^N$, $ f(x)\leq1$ for all $ x\in\mathbb{R}^N$, $ 1<p<+\infty$ for $ N=1,2$, $ 1<p<\frac{N+2}{N-2}$ for $ N\geq3$. We prove that for any $ k=1,2,\,\ldots\,$, if $ \lambda$ is large enough the above problem has positive solutions $ u_\lambda$ concentrating at $ k$ distinct points away from the origin as $ \lambda$ goes to $ \infty$.

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Additional Information:

Daomin Cao
Affiliation: Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, People's Republic of China
Email: dmcao@amt.ac.cn

Ezzat S. Noussair
Affiliation: School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Email: noussair@maths.unsw.edu.au

Shusen Yan
Affiliation: School of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW 2351, Australia
Email: syan@turing.une.edu.au

PII: S0033-569X-06-01026-5
Keywords: Nonlinear elliptic equation, asymmetric positive solutions, variational method, critical point
Received by editor(s): August 2, 2005
Posted: June 21, 2006
Additional Notes: The first author was supported by the Funds of Distinguished Young Scholars of NSFC and Innovation Funds of CAS in China
The second author was supported by ARC in Australia
The third author was supported by ARC in Australia
Copyright of article: Copyright 2006, Brown University


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