Quasi-radial nodal solutions for the Lane-Emden problem in the ball

Abstract

We consider the semilinear elliptic problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B\\ u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^2$ centered at the origin and $p\in (1,+\infty)$. We prove the existence of non-radial sign-changing solutions to \eqref{problemAbstract} which are \emph{quasi-radial}, namely solutions whose nodal line is the union of a finite number of disjoint simple closed curves, which are the boundary of nested domains contained in $B$. In particular the nodal line of these solutions doesn't touch $\partial B$. \\ The result is obtained with two different approaches: via nonradial bifurcation from the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} at certain values of $p$ and by investigating the qualitative properties, for $p$ large, of the least energy nodal solutions in spaces of functions invariant by the action of the dihedral group generated by the reflection with respect to the $x$-axis and the rotation about the origin of angle $\frac{2\pi}{k}$ for suitable integers $k$.\\ We also prove that for certain integers $k$ the least energy nodal solutions in these spaces of symmetric functions are instead radial, showing in particular a breaking of symmetry phenomenon in dependence on the exponent $p$.