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).


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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