Symmetry of solutions to semilinear elliptic equations via Morse index

Pacella, Filomena; Weth, Tobias

doi:10.1090/s0002-9939-07-08652-2

Cited by 60 publications

(63 citation statements)

References 15 publications

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“…Nevertheless in some situations, or for a certain class of solutions, it is natural to expect that solutions inherit some of the symmetry of the domain, even if Ω is not convex, u changes sign and f is increasing in the radial variable. This is indeed the case for solutions of low Morse index, under some convexity assumption on f that we shall make clear very soon, and it has been proved in [30,31] when Ω is bounded, and in [27] when Ω is unbounded, that every solution to (1.1) (and possibly (1.2)) of low Morse index is axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis, i.e. only depends on r = |x| and θ = arccos(x/|x| • p), for a certain unit vector p, and u is nonincreasing in θ.…”

Section: Introductionmentioning

confidence: 58%

“…So also nodal solutions can be considered. According to the previous results in [27,31] we believe moreover that this assumption should be optimal and that allowing a higher k-Morse index would produce more symmetries in the sectors S 2k,e .…”

Section: Introductionmentioning

confidence: 73%

“…In this paper, we are interested in solutions which admit some invariances, namely they belong to suitable symmetric spaces, in the case when Ω is contained in the plane. Inspiring to the previous depicted papers [27,31] we consider solutions which have low Morse index in these symmetric spaces and we prove, that under some convexity assumptions on f or they are radial or they inherit only one extra-symmetry among the ones they can possess. This extra-symmetry of low Morse index solutions is what we think the right generalization to the foliated Schwarz symmetry to this symmetric setting.…”

Section: Introductionmentioning

confidence: 84%

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A monotonicity result under symmetry and Morse index constraints in the plane

Gladiali

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

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Section: Introductionmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 84%

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A monotonicity result under symmetry and Morse index constraints in the plane

Gladiali

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…The relevant properties of polarization are that it approximates the symmetrization and at the same time is continuous in W 1,2 . The use of polarizations to prove symmetry of solutions of partial differential equations has appeared in a number of works (see [8,40,41,45]).…”

Section: Symmetrization In the Spherementioning

confidence: 99%

Ground states of semilinear elliptic problems with applications to the Allen–Cahn equation on the sphere

et al. 2022

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We study solutions of $$\Delta u - F'(u)=0$$ Δ u - F ′ ( u ) = 0 , where the potential F can have an arbitrary number of wells at arbitrary heights, including bottomless wells with subcritical decay. In our setting, ground state solutions correspond to unstable solutions of least energy. We show that in convex domains of $${\mathbb {R}}^N$$ R N and manifolds with $${\text {Ric}}\ge 0$$ Ric ≥ 0 , ground states are always of mountain-pass type and have Morse index 1. In addition, we prove symmetry of the ground states if the domain is either an Euclidean ball or the entire sphere $$S^{N}$$ S N . For the Allen–Cahn equation $$\varepsilon ^2\Delta u - W'(u)=0$$ ε 2 Δ u - W ′ ( u ) = 0 on $$S^{N}$$ S N , we prove the ground state is unique up to rotations and corresponds to the equator as a minimal hypersurface. We also study bifurcation at the energy level of the ground state as $$\varepsilon \rightarrow 0$$ ε → 0 , showing that the first $$N+1$$ N + 1 min-max Allen–Cahn widths of $$S^{N}$$ S N are ground states, and we prove a gap theorem for the corresponding $$(N+2)$$ ( N + 2 ) -th min-max solution.

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“…For related results, we refer to [1], [4]- [10], [12], [13], [17]- [20], [22]- [27], [29], [30]. For recent developments of the symmetry of positive solutions of (1.1), see [2], [3], [15], [28], [33] and references therein.…”

Section: Introductionmentioning

confidence: 99%