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

@article{Amadori2018OnAS,
  title={On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s: II},
  author={Anna Lisa Amadori and Francesca Gladiali},
  journal={Nonlinearity},
  year={2018},
  volume={33},
  pages={2541 - 2561}
}
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 −Δu=|x|αf(u)inΩ,u=0on∂Ω, where Ω 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… 
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References

SHOWING 1-10 OF 83 REFERENCES

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Ω ⊂ R be a ball or an annulus and f : R → R absolutely continuous, superlinear, subcritical, and such that f(0) = 0. We prove that the least energy nodal solution of −∆u = f(u), u ∈ H 0 (Ω), is…

Morse index of radial nodal solutions of Henon type equations in dimension two

We consider non-autonomous semilinear elliptic equations of the type −Δu = |x|αf(u),x ∈ Ω,u =0on∂Ω, where Ω ⊂ ℝ2 is either a ball or an annulus centered at the origin, α > 0 and f : ℝ → ℝ is C1,β on…

Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

SYMMETRY BREAKING FOR A CLASS OF SEMILINEAR ELLIPTIC PROBLEMS

We study positive solutions of the Dirichlet problem for —Aji = up — A, p > 1, A > 0, on the unit ball 0. We show that there exists a positive solution (uo, Ao) of this problem which satisfies in…

Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane

In this paper, we study the problem −Δu = 2+α 22|x|αf(λ,u),in B 1, u > 0, in B1, u = 0, on ∂B1, (𝒫) where B1 is the unit ball of ℝ2, f is a smooth nonlinearity and α, λ are real numbers with α > 0.…

Quasilinear elliptic equations with critical exponents

has no solution if Ω ⊂ R , N ≥ 3, is bounded and starshaped with respect to some point, and 2∗ = 2N/(N − 2). In (P0) the nonlinear term is a power of u with the critical exponent (N + 2)/(N − 2).…

Lane Emden problems with large exponents and singular Liouville equations

Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains

Quasi-radial solutions for the Lane–Emden problem in the ball

We consider the semilinear elliptic problem where B is the unit ball of $${\mathbb {R}}^2$$ R 2 centered at the origin and $$p\in (1,+\infty )$$ p ∈ ( 1 , + ∞ ) . We prove the existence of…
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