On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's: II
Abstract
By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in Amadori A L and Gladiali F (2018 arXiv:1805.04321), we give a lower bound for the Morse index of radial solutions to Hénon type problems $\begin{cases}-{\Delta}u={\vert x\vert }^{\alpha }f\left(u\right)\quad \quad \quad \hfill & \text{in}\;\;\;\;\;{\Omega},\hfill \\ u=0\quad \hfill & \text{on}\;\partial {\Omega},\hfill \end{cases}$ where Ω is a bounded radially symmetric domain of ${\mathbb{R}}^{N}$ (N ⩾ 2), α > 0 and f is a real function. From this estimate we get that the Morse index of nodal radial solutions to this problem goes to ∞ as α → ∞. Concerning the real Hénon problem, f(u) = |u|p-1u, we prove radial nondegeneracy, we show that the radial Morse index is equal to the number of nodal zones and we get that a least energy nodal solution is not radial. *This work was supported by Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
- Publication:
-
Nonlinearity
- Pub Date:
- June 2020
- DOI:
- 10.1088/1361-6544/ab7639
- arXiv:
- arXiv:1906.00368
- Bibcode:
- 2020Nonli..33.2541A
- Keywords:
-
- semilinear elliptic equations;
- nodal solutions;
- Morse index;
- radial solutions;
- Hénon type problems;
- 35J25;
- 35J61;
- 35P15;
- 35B05;
- 35B06;
- 35A24;
- Mathematics - Analysis of PDEs
- E-Print:
- doi:10.1088/1361-6544/ab7639