Anisotropic elliptic equations with general growth in the gradient and Hardy-type potentials

Pietra, Francesco Della; Gavitone, Nunzia

doi:10.1016/j.jde.2013.07.019

Cited by 15 publications

(9 citation statements)

References 30 publications

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“…Many other authors developed the related theory in several settings, considering both analytic aspects (see [4,7,8,11,[13][14][15][16]27,28]) and geometric points of view (see [9,12,19]). In this paper we will study the anisotropic capacity problem (6) and the associated overdetermined problem (6)- (7).…”

Section: Resultsmentioning

confidence: 99%

“…where V = H 2 /2 and the last equality holds thanks to the homogeneity of V (·), which follows from (15). Recalling the definition of H-capacity, the fact that div(∇ ξ V (Du)) = 0 in R N \ and that ν = −Du/|Du| on ∂ , and by using the homogeneity of V , the latter can be rewritten as…”

Section: Proof Of Theorem 11mentioning

confidence: 97%

“…When H and H 0 are of class C 2 (R N \{0}), by differentiating this expression and using (15) and (16), we obtain…”

Section: Norms In Rmentioning

confidence: 99%

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An overdetermined problem for the anisotropic capacity

2016

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We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in RN, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm H0)

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Section: Resultsmentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 97%

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An overdetermined problem for the anisotropic capacity

2016

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“…Moreover, we study the behavior of the blow-up solutions of (1.1) when λ → 0 + . Problems which deal with Finsler-Laplacian type operators have been studied in several contexts (see, for example, [AFTL,BFK,FK,CS,WX,CFV,DG1,DG2,DG3,Ja]).…”

Section: Introductionmentioning

confidence: 99%

Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

Pietra

Blasio

2017

Publicacions Matemàtiques

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In this paper we prove existence results and asymptotic behavior for strong solutions u ∈ W 2,2 loc (Ω) of the nonlinear elliptic problem (P) −∆ H u + H(∇u) q + λu = f in Ω, u → +∞ on ∂Ω, where H is a suitable norm of R n , Ω ⊂ R n is a bounded domain, ∆ H is the Finsler Laplacian, 1 < q ≤ 2, λ > 0, and f is a suitable function in L ∞ loc. Furthermore, we are interested in the behavior of the solutions when λ → 0 + , studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P).

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“…This kind of operator has been studied in several papers (see for instance , , , , for

p = 2

, and , , , for

1 < p < \infty)

.…”

Section: Introductionmentioning

confidence: 99%

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

Pietra

Gavitone

2013

Mathematische Nachrichten

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In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator Δpu=∑i∂∂xi|∇u|p−2-0.16em∂u∂xi, that is the p‐Laplacian, and trueΔ̃pu=∑i∂∂xi|∂u∂xi|p−2∂u∂xi, namely the pseudo‐p‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p‐torsional rigidity.

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Anisotropic elliptic equations with general growth in the gradient and Hardy-type potentials

Cited by 15 publications

References 30 publications

An overdetermined problem for the anisotropic capacity

An overdetermined problem for the anisotropic capacity

Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

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