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Infinitely many solutions of a symmetric Dirichlet problem. (English) Zbl 0799.35071
We seek solutions \(u=(u_ 1,\dots, u_ m): \overline{\Omega} \to \mathbb{R}^ m\) of the nonlinear Dirichlet problem \[ \Delta u+F_ u(u)=u \quad \text{in }\Omega, \qquad u=0 \quad \text{on } \partial\Omega. \tag{D} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) with smooth boundary and \(F: \mathbb{R}^ m \to\mathbb{R}\) is \(C^ 1\) and satisfies certain growth conditions.
We investigate which kind of symmetries guarantee the existence of infinitely many solutions of (D).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
58J70 Invariance and symmetry properties for PDEs on manifolds
35J20 Variational methods for second-order elliptic equations
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