On a nonlinear elliptic equation arising in a free boundary problem

Guofang Wang; Dong Ye

doi:10.1007/s00209-003-0493-3

On a nonlinear elliptic equation arising in a free boundary problem

Guofang Wang^*
, Dong Ye

^*Corresponding author for this work

Max Planck Institute for Mathematics in the Sciences
CY Cergy Paris Université
Ecole Normale Supérieure Paris Saclay

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

Let p* = n/(n -2) and n ≥ 3. In this paper, we first classify all non-constant solutions of { -Δu = u₊^p* in ℝⁿ, ∫_ℝnu₊^p*dx < ∞. We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation -Δu = u₊^p* . Our results illustrate that this equation is much closer to the Liouville problem -Δu = e^u in dimension two than the usual critical exponent equation, namely -Δu = u^n+2/n-2 is.

Original language	English
Pages (from-to)	531-548
Number of pages	18
Journal	Mathematische Zeitschrift
Volume	244
Issue number	3
DOIs	https://doi.org/10.1007/s00209-003-0493-3
State	Published - Jul 2003
Externally published	Yes

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title = "On a nonlinear elliptic equation arising in a free boundary problem",

abstract = "Let p* = n/(n -2) and n ≥ 3. In this paper, we first classify all non-constant solutions of \{ -Δu = u+p* in ℝn, ∫ℝnu+p*dx < ∞. We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation -Δu = u+p* . Our results illustrate that this equation is much closer to the Liouville problem -Δu = eu in dimension two than the usual critical exponent equation, namely -Δu = un+2/n-2 is.",

author = "Guofang Wang and Dong Ye",

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doi = "10.1007/s00209-003-0493-3",

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AU - Wang, Guofang

AU - Ye, Dong

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N2 - Let p* = n/(n -2) and n ≥ 3. In this paper, we first classify all non-constant solutions of { -Δu = u+p* in ℝn, ∫ℝnu+p*dx < ∞. We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation -Δu = u+p* . Our results illustrate that this equation is much closer to the Liouville problem -Δu = eu in dimension two than the usual critical exponent equation, namely -Δu = un+2/n-2 is.

AB - Let p* = n/(n -2) and n ≥ 3. In this paper, we first classify all non-constant solutions of { -Δu = u+p* in ℝn, ∫ℝnu+p*dx < ∞. We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation -Δu = u+p* . Our results illustrate that this equation is much closer to the Liouville problem -Δu = eu in dimension two than the usual critical exponent equation, namely -Δu = un+2/n-2 is.

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VL - 244

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JO - Mathematische Zeitschrift

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ER -

On a nonlinear elliptic equation arising in a free boundary problem

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