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Strongly indefinite functionals and multiple solutions of elliptic systems. (English) Zbl 1125.35338

Summary: We study existence and multiplicity of solutions of the elliptic system \[ \begin{cases} -\Delta u =H_u(x,u,v) & \text{ in } \Omega,\\ -\Delta v =-H_v(x,u,v) \text{ in }, \quad u(x) =v(x)=0 \text{ on } \partial \Omega ,\end{cases} \] 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 \[ H(x,u,v)\sim | u|^p + | v|^q + R(x,u,v) \;\text{ with }\lim_{|(u,v)| \to\infty} \frac{R(x,u,v)}{| u|^p+| v|^q}=0, \] 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.

MSC:

35J57 Boundary value problems for second-order elliptic systems
35J50 Variational methods for elliptic systems
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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