Kelvin transform for Grushin operators and critical semilinear equations

Monti, Roberto; Morbidelli, Daniele

doi:10.1215/s0012-7094-05-13115-5

Cited by 62 publications

(55 citation statements)

References 19 publications

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“…In this subsection we show that inversion is conformal. The same result has been proved in [18], but here we provide a shorter proof, using the warped model. Let Φ(z) = δ z −2 z.…”

Section: 2supporting

confidence: 77%

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Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

Morbidelli

2009

Isr. J. Math.

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We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields in R n . It turns out that in many cases all such maps can be obtained as compositions of suitable dilations, inversions and isometries. Our methods involve a study of the singular Riemannian metric associated with the vector fields. In particular, we identify some conformally invariant cones related to the Weyl tensor. The knowledge of such cones enables us to classify all umbilical hypersurfaces.

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“…In this subsection we show that inversion is conformal. The same result has been proved in [18], but here we provide a shorter proof, using the warped model. Let Φ(z) = δ z −2 z.…”

Section: 2supporting

confidence: 77%

“…See the discussion in Subsection 2.2. The conformality of the map Φ was already recognized in [18] by R. Monti for all z = (x, y) ∈ Ω. Here Γ is an isometry of the form (1.3), t > 0, b ∈ R q and s = 0 or −2.…”

Section: Introductionmentioning

confidence: 83%

Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

Morbidelli

2009

Isr. J. Math.

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“…The Grushin operator G γ is not invariant with respect to translations and reflections about hyperplanes in all the directions of R d+k , and hence, as remarked by MontiMorbidelli [20], it is not possible to apply the technique of moving planes to this operator.…”

Section: Introductionmentioning

confidence: 99%

“…In the paper of Monti-Morbidelli [20] it is also shown that any solution of problem (2) with p = (Q + 2)/(Q − 2) must exhibit a kind of "spherical symmetry", by exploiting the invariance of the equation with respect to a suitable conformal inversion, i.e. the Kelvin transform for the Grushin operator (see also Lupo-Payne [18] for further details).…”

Section: Introductionmentioning

confidence: 99%

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Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Monticelli¹

2010

J. Eur. Math. Soc.

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Abstract. We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation. A good example of such an operator is the Grushin operator on R d+k , to which we devote particular attention. As an application of these tools in the degenerate elliptic setting, we prove a partial symmetry result for classical solutions of semilinear problems on bounded, symmetric and suitably convex domains, which is a generalization of the result of Gidas-Ni-Nirenberg [12], [13], and a nonexistence result for classical solutions of semilinear equations with subcritical growth defined on the whole space, which is a generalization of the result of Gidas-Spruck [14] and Chen-Li [6]. We use the method of moving planes, implemented just in the directions parallel to the degeneracy set of the Grushin operator.

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Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

Alves,

de Holanda

2023

Mathematische Nachrichten

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In this work, we study the existence of the solution for a class of semilinear degenerate elliptic equations in the whole space involving the Grushin operator and a nonlinearity that involves the Grushin's critical growth and the exponential growth. The existence of at least one nontrivial weak solution is done by combining the mountain‐pass theorem, Trudinger–Moser inequality, and a version of a result due to Lions for the exponential growth in in the corresponding weighted Sobolev space.

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Kelvin transform for Grushin operators and critical semilinear equations

Cited by 62 publications

References 19 publications

Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators

Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

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