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Zbl 0799.35071
Bartsch, Thomas
Infinitely many solutions of a symmetric Dirichlet problem.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 20, No.10, 1205-1216 (1993). ISSN 0362-546X

We seek solutions $u=(u\sb 1,\dots, u\sb m): \overline{\Omega} \to \bbfR\sp m$ of the nonlinear Dirichlet problem $$\Delta u+F\sb u(u)=u \quad \text {in }\Omega, \qquad u=0 \quad \text {on } \partial\Omega. \tag D$$ Here $\Omega$ is a bounded domain in $\bbfR\sp n$ with smooth boundary and $F: \bbfR\sp m \to\bbfR$ is $C\sp 1$ and satisfies certain growth conditions.\par We investigate which kind of symmetries guarantee the existence of infinitely many solutions of (D).
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
58J70 Invariance and symmetry properties
35J20 Second order elliptic equations, variational methods

Keywords: Borsuk-Ulam theorem; geometrical index theory; symmetries; infinitely many solutions

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