### Non-radial solutions with orthogonal subgroup invariance for semilinear Dirichlet problems

#### Abstract

A semilinear elliptic equation, $-\Delta u=\lambda f(u)$, is studied in a ball

with the Dirichlet boundary condition. For a closed subgroup $G$ of the

orthogonal group, it is proved that the number of non-radial $G$ invariant solutions

diverges to infinity as $\lambda$ tends to $\infty$ if $G$

is not transitive on the unit sphere.

with the Dirichlet boundary condition. For a closed subgroup $G$ of the

orthogonal group, it is proved that the number of non-radial $G$ invariant solutions

diverges to infinity as $\lambda$ tends to $\infty$ if $G$

is not transitive on the unit sphere.

#### Keywords

Semilinear elliptic equation; group invariant solution; non-radial solution; variational method

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