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-&Deltau= xα f(u)inΩ, u=0 onΩ, is a bounded radially symmetric domain of RN (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-1 u, 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: II / Amadori, A. L.; Gladiali, F.. - In: NONLINEARITY. - ISSN 0951-7715. - 33:6(2020), pp. 2541-2561. [10.1088/1361-6544/ab7639]

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

Gladiali F.
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 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-&Deltau= xα f(u)inΩ, u=0 onΩ, is a bounded radially symmetric domain of RN (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-1 u, 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.
2020
On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's: II / Amadori, A. L.; Gladiali, F.. - In: NONLINEARITY. - ISSN 0951-7715. - 33:6(2020), pp. 2541-2561. [10.1088/1361-6544/ab7639]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11388/240786
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