By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in a previous paper, we give a lower bound for the Morse index of radial solutions to H'enon type problems [ left{egin{array}{ll} -Delta u = |x|^alpha f(u) qquad & ext{ in } Omega, u= 0 & ext{ on } partial Omega, end{array} ight. ] where $Omega$ is a bounded radially symmetric domain of $mathbb R^N$ ($Nge 2$), $alpha>0$ and $f$ is a real function. From this estimate we get that the Morse index of nodal radial solutions to this problem goes to $infty$ as $alpha o infty$. Concerning the real H'enon 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.

On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's, Part II

Anna Lisa Amadori;
2020-01-01

Abstract

By using a characterization of the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem given in a previous paper, we give a lower bound for the Morse index of radial solutions to H'enon type problems [ left{egin{array}{ll} -Delta u = |x|^alpha f(u) qquad & ext{ in } Omega, u= 0 & ext{ on } partial Omega, end{array} ight. ] where $Omega$ is a bounded radially symmetric domain of $mathbb R^N$ ($Nge 2$), $alpha>0$ and $f$ is a real function. From this estimate we get that the Morse index of nodal radial solutions to this problem goes to $infty$ as $alpha o infty$. Concerning the real H'enon 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11367/76475
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