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Non-radial solutions with orthogonal subgroup invariance for semilinear Dirichlet problems. (English) Zbl 1046.35031

Let \(\Omega = \{x\in {\mathbb R}^n : | x| <R\}\) and let \(G\) be a closed subgroup of \(O(n)\). \(G\) is called transitive if for each \(x,y\in S^{n-1}\) there exists \(g\in G\) such that \(gx=y\). The author studies the existence of non-radial \(G\)-invariant solutions of the boundary value problem \(-\Delta u = \lambda f(u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\). Here \(\lambda>0\) is a parameter and two typical examples of \(f\) are \(f(u) = u-| u| ^{p-1}u\), where \(p>1\), and \(f(u) = \sin u\).
The main result of the paper asserts that \(G\) is non-transitive if and only if for each positive integer \(k\) there exists \(\lambda_{k}\) such that whenever \(\lambda > \lambda_{k}\), then the boundary value problem above has at least \(k\) non-radial \(G\)-invariant solutions. The proof uses Clark’s theorem in critical point theory for even functionals. More precisely, solutions are found as critical points of the Euler-Lagrange functional in the Sobolev space \(H^1_{0,G}(\Omega)\) of \(G\)-invariant functions, and existence of non-radial solutions is established by comparing the total number of critical points with the number of critical points corresponding to the radial solutions. The paper is a continuation of the author’s earlier work where a similar problem, but for a different class of nonlinearities, has been considered.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
47J30 Variational methods involving nonlinear operators
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