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- W2756312437 abstract "We consider the semilinear elliptic problem begin{equation}label{problemAbstract} left{begin{array}{lr}-Delta u= |u|^{p-1}uqquad mbox{ in }B u=0qquadqquadqquadmbox{ 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 $pin (1,+infty)$. 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 $frac{2pi}{k}$ 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 $p$." @default.
- W2756312437 created "2017-09-25" @default.
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- W2756312437 date "2017-09-11" @default.
- W2756312437 modified "2023-10-16" @default.
- W2756312437 title "Quasi-radial nodal solutions for the Lane-Emden problem in the ball" @default.
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