## 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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