| 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} having $2$ nodal domains and whose nodal line does not 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 $p$. |
|