Stable solutions of $−\Delta u = f(u)$ in $\mathbb{R}^N$

Dupaigne, Louis; Farina, Alberto

doi:10.4171/jems/217

Cited by 77 publications

(60 citation statements)

References 16 publications

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“…More recently, in [10], Liouville type theorems for stable solutions were obtained for a rather general class of semilinear equations, and in [9], Farina's idea was applied to semilinear problems whose nonlinearity has a negative power to obtain various qualitative properties of finite Morse index solutions. Further related results may be found in [28,19,20,11] and the references therein.…”

Section: Introductionmentioning

confidence: 89%

Finite Morse index solutions of an elliptic equation with supercritical exponent

Dancer

Zhang

2011

Journal of Differential Equations

102

105

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We study the behavior of finite Morse index solutions of the equationwhere p > 1, α > −2, and Ω is a bounded or unbounded domain.We show that there is a critical power p = p(α) larger than the usual critical exponent N+2 N−2 such that this equation with Ω = R N has no nontrivial stable solution for 1 < p < p(α) but it admits a family of stable positive solutions when p p(α). For a positive solution u with finite Morse index, we classify the singularity of u at the origin when Ω is a punctured ball B R (0)\{0}, and we classify its behavior at infinity when Ω = R N \ B R (0). We show that the behavior depends crucially on whether p is below or above the critical power p(α). We also demonstrate how a duality method can be used to obtain sharper results. ✩ We thank Alberto Farina for pointing out a mistake in an earlier version of the paper. Z.M. Guo would like to thank the University of Sydney for a nice stay.

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Section: Introductionmentioning

confidence: 89%

Finite Morse index solutions of an elliptic equation with supercritical exponent

Dancer

Zhang

2011

Journal of Differential Equations

102

105

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“…In an elegant paper, Farina [11] obtained a sharp classification for all finite Morse indices solutions of (1.3) (see also [10]). …”

Section: Introductionmentioning

confidence: 99%

Morse indices of solutions for super-linear elliptic PDE’s

Hajlaoui

Harrabi

Mtiri

2015

Nonlinear Analysis: Theory, Methods & Applications

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“…As it is shown by Dupaigne and Farina in [14], any classical bounded stable solution of (1) is constant provided 1 ≤ n ≤ 4 and 0 ≤ H ∈ C 1 (R) is a general nonlinearity. For particular nonlinearities H(u) = e u and H(u) = u p where p > 1 the differential equation (1) is called Gelfand and Lane-Emden equations, respectively, and optimal Liouville theorems are provided by Farina in [18,19].…”

Section: Introductionmentioning

confidence: 93%

Rigidity results for stable solutions of symmetric systems

Fazly

2015

Proc. Amer. Math. Soc.

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Abstract. We study stable solutions of the following nonlinear systemand Ω is a domain in R n . We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of H is symmetric. It seems that this concept is crucial to prove Liouville theorems, when Ω = R n , and regularity results, when Ω = B 1 , for stable solutions of the above system for a general nonlinearity H ∈ C 1 (R m ). Moreover, we provide an improvement for a linear Liouville theorem given in [20] that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.

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Stable solutions of $−\Delta u = f(u)$ in $\mathbb{R}^N$

Abstract: Abstract. Several Liouville-type theorems are presented for stable solutions of the equation − u = f (u) in R N , where f > 0 is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

Cited by 77 publications

References 16 publications

Finite Morse index solutions of an elliptic equation with supercritical exponent

Finite Morse index solutions of an elliptic equation with supercritical exponent

Morse indices of solutions for super-linear elliptic PDE’s

Rigidity results for stable solutions of symmetric systems

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