Exact Morse index computation for nodal radial solutions of Lane-Emden problems

Abstract

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-Δu= |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 ${\mathbb{R}}^N$, $N\ge 2$, centered at the origin and $1<p<p_S$, with $p_S=+\infty$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\ge 3$. Our main result is to prove that in dimension $N=2$ the Morse index of the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} is exactly $12$ if $p$ is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in ${\mathbb{R}}^N$ in any dimension $N\ge 2$.

References in corpus (1)

• Morse index and sign changing bubble towers for Lane-Emden problems

Cited by in corpus (8)

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• Nonradial sign changing solutions to Lane Emden equation
• The Hénon problem with large exponent in the disc
• On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's, Part II
• Quasi-radial nodal solutions for the Lane-Emden problem in the ball
• Global bifurcation for the Hénon problem
• Monotonicity of the Morse index of radial solutions of the Hénon equation in dimension two
• Morse Index of Multiple Blow-Up Solutions to the Two-Dimensional Sinh-Poisson Equation