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Transactions of the American Mathematical Society

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



Strongly indefinite functionals and multiple solutions of elliptic systems

Authors: D. G. De Figueiredo and Y. H. Ding
Journal: Trans. Amer. Math. Soc. 355 (2003), 2973-2989
MSC (2000): Primary 35J50; Secondary 58E99
Published electronically: March 14, 2003
MathSciNet review: 1975408
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Abstract: We study existence and multiplicity of solutions of the elliptic system

\begin{displaymath}\begin{cases}-\Delta u =H_u(x,u,v) & \text{in} \Omega,\\ -\D... ...\quad u(x) =v(x)=0 \quad\text{on} \partial \Omega ,\end{cases}\end{displaymath}

where $\Omega\subset\mathbb{R}^N, N\geq 3$, is a smooth bounded domain and $H\in \mathcal{C}^1(\bar{\Omega}\times\mathbb{R}^2, \mathbb{R})$. We assume that the nonlinear term

\begin{displaymath}H(x,u,v)\sim \vert u\vert^p + \vert v\vert^q + R(x,u,v) \ \te... ...ert\to\infty}\frac{R(x,u,v)}{\vert u\vert^p+\vert v\vert^q}=0, \end{displaymath}

where $p\in (1, 2^*)$, $2^*:=2N/(N-2)$, and $q\in (1, \ \infty)$. So some supercritical systems are included. Nontrivial solutions are obtained. When $H(x,u,v)$ is even in $(u,v)$, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if $p>2$ (resp. $p<2$). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

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Additional Information

D. G. De Figueiredo
Affiliation: IMECC-UNICAMP, Caixa Postal 6065, 13083-970 Campinas S.P. Brazil

Y. H. Ding
Affiliation: Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100080 Beijing, People’s Republic of China

Keywords: Elliptic system, multiple solutions, critical point theory
Received by editor(s): June 18, 2001
Published electronically: March 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society