Energy and Morse index of solutions of Yamabe type problems on thin annuli

AYED, M; Mehdi, Khalil El; Ahmedou, Mohameden Ould; Pacella, Filomena

doi:10.4171/jems/29

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…The fact that we are not able to construct the solutions for all values of ǫ close enough to 0 is also reflected in another important feature of our solutions, namely that their Morse index tends to infinity as ǫ tends to 0. In the same way, recall that when p = m+2 m−2 and k = m − 1 it has been proved in [4] that the energy and the Morse index of all positive solutions tend to infinity as ǫ tends to 0.…”

Section: Introductionmentioning

confidence: 78%

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Solutions of Semilinear Elliptic Equations in Tubes

2012

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Given a smooth compact k-dimensional manifold Λ embedded in ℝ m, with m≥2 and 1≤k≤m-1, and given ε{lunate}>0, we define B ε{lunate}(Λ) to be the geodesic tubular neighborhood of radius ε{lunate} about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation {Mathematical expression} when the parameter ε{lunate} is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m-k≤2 or {Mathematical expression} when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ε{lunate} tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for {Mathematical expression}, n≥3, if ε{lunate} is sufficiently small. © 2012 Mathematica Josephina, Inc

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Section: Introductionmentioning

confidence: 78%

“…In the same way, recall that when p = m+2 m−2 and k = m − 1 it has been proved in [4] that the energy and the Morse index of all positive solutions tend to infinity as ǫ tends to 0.…”

Section: Introductionmentioning

confidence: 78%

Solutions of Semilinear Elliptic Equations in Tubes

2012

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Exact Morse index of radial solutions for semilinear elliptic equations with critical exponent on annuli

Miyamoto

2023

Math. Z.

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Let $$N\ge 3$$ N ≥ 3 , $$R>\rho >0$$ R > ρ > 0 and $$A_{\rho }:=\{x\in \mathbb {R}^N;\ \rho<|x|<R\}$$ A ρ : = { x ∈ R N ; ρ < | x | < R } . Let $$U^{\pm }_{n,\rho }$$ U n , ρ ± , $$n\ge 1$$ n ≥ 1 , be a radial solution with n nodal domains of $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta U+|x|^{\alpha }|U|^{p-1}U=0 &{} \text {in}\ A_{\rho },\\ U=0 &{} \text {on}\ \partial A_{\rho }. \end{array}\right. } \end{aligned}$$ Δ U + | x | α | U | p - 1 U = 0 in A ρ , U = 0 on ∂ A ρ . We show that if $$p=\frac{N+2+2\alpha }{N-2}$$ p = N + 2 + 2 α N - 2 , $$\alpha >-2$$ α > - 2 and $$N\ge 3$$ N ≥ 3 , then $$U^{\pm }_{n,\rho }$$ U n , ρ ± is nondegenerate for small $$\rho >0$$ ρ > 0 and the Morse index $${{\textsf{m}}}(U^{\pm }_{n,\rho })$$ m ( U n , ρ ± ) satisfies $$\begin{aligned} {{\textsf{m}}}(U^{\pm }_{n,\rho }) =n\frac{(N+2\ell -1)(N+\ell -1)!}{(N-1)!\ell !} \quad \text {for small}\ \rho >0, \end{aligned}$$ m ( U n , ρ ± ) = n ( N + 2 ℓ - 1 ) ( N + ℓ - 1 ) ! ( N - 1 ) ! ℓ ! for small ρ > 0 , where $$\ell =[\frac{\alpha }{2}]+1$$ ℓ = [ α 2 ] + 1 . Using Jacobi elliptic functions, we show that if $$(p,\alpha )=(3,N-4)$$ ( p , α ) = ( 3 , N - 4 ) and $$N\ge 3$$ N ≥ 3 , then the Morse index of a positive and negative solutions $${{\textsf{m}}}(U^{\pm }_{1,\rho })$$ m ( U 1 , ρ ± ) is completely determined by the ratio $$\rho /R\in (0,1)$$ ρ / R ∈ ( 0 , 1 ) . Upper and lower bounds for $${{\textsf{m}}}(U^{\pm }_{n,\rho })$$ m ( U n , ρ ± ) , $$n\ge 1$$ n ≥ 1 , are also obtained when $$(p,\alpha )=(3,N-4)$$ ( p , α ) = ( 3 , N - 4 ) and $$N\ge 3$$ N ≥ 3 .

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Characterization of solitons in a pseudo-quasi-conformally flat and pseudo- $ W_8 $ flat Lorentzian Kähler space-time manifolds

Chaturvedi,

Kaushik,

Bhagat

et al. 2024

MATH

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<abstract><p>The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-$ W_8 $ flat Lorentzian Kähler space-time manifolds.</p></abstract>

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Energy and Morse index of solutions of Yamabe type problems on thin annuli

Cited by 3 publications

References 21 publications

Solutions of Semilinear Elliptic Equations in Tubes

Solutions of Semilinear Elliptic Equations in Tubes

Exact Morse index of radial solutions for semilinear elliptic equations with critical exponent on annuli

Characterization of solitons in a pseudo-quasi-conformally flat and pseudo- $ W_8 $ flat Lorentzian Kähler space-time manifolds

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