Energy asymptotics in the Brezis–Nirenberg problem: The higher-dimensional case

Rupert L. Frank; Tobias König; Hynek Kovařík; Rupert L. Frank; Tobias König; Hynek Kovařík

doi:10.3934/mine.2020007

Mathematics in Engineering

2020, Volume 2, Issue 1: 119-140. doi: 10.3934/mine.2020007

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Energy asymptotics in the Brezis–Nirenberg problem: The higher-dimensional case

1.
Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany
2.
Mathematics 253-37, Caltech, Pasadena, CA 91125, USA
3.
DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 38-25123 Brescia, Italy

Received: 24 October 2019 Accepted: 23 November 2019 Published: 06 December 2019

For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding $ S(\epsilon V) : = \inf\limits_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega |\nabla u|^2\, dx +\epsilon \int_\Omega V\, |u|^2\, dx}{\left(\int_\Omega |u|^q \, dx \right)^{2/q}}, $ where $q = \frac{2N}{N-2}$ is the critical Sobolev exponent, $\Omega \subset \mathbb{R}^N$ is a bounded open set and $V:\overline{\Omega}\to \mathbb{R}$ is a continuous function. We compute the asymptotics of $S(0) - S(\epsilon V)$ to leading order as $\epsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case $N = 3$.
- Brezis–Nirenberg problem,
- energy asymptotic,
- minimizing sequences,
- blow-up
Citation: Rupert L. Frank, Tobias König, Hynek Kovařík. Energy asymptotics in the Brezis–Nirenberg problem: The higher-dimensional case[J]. Mathematics in Engineering, 2020, 2(1): 119-140. doi: 10.3934/mine.2020007

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Abstract

For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding $ S(\epsilon V) : = \inf\limits_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega |\nabla u|^2\, dx +\epsilon \int_\Omega V\, |u|^2\, dx}{\left(\int_\Omega |u|^q \, dx \right)^{2/q}}, $ where $q = \frac{2N}{N-2}$ is the critical Sobolev exponent, $\Omega \subset \mathbb{R}^N$ is a bounded open set and $V:\overline{\Omega}\to \mathbb{R}$ is a continuous function. We compute the asymptotics of $S(0) - S(\epsilon V)$ to leading order as $\epsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case $N = 3$.

References

[1]	Aubin T (1976) Problèmes isopérimétriques et espaces de Sobolev. J Differ Geom 11: 573-598. doi: 10.4310/jdg/1214433725
[2]	Bahri A, Coron JM (1988) On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain. Commun Pur Appl Math 41: 253-294. doi: 10.1002/cpa.3160410302
[3]	Brézis H, Nirenberg L (1983) Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun Pur Appl Math 36: 437-477. doi: 10.1002/cpa.3160360405
[4]	Brézis H, Peletier LA (1989) Asymptotics for elliptic equations involving critical growth, In: Partial Differential Equations and the Calculus of Variations, Boston: Birkhäuser, 149-192.
[5]	Flucher M, Wei J (1998) Asymptotic shape and location of small cores in elliptic free-boundary problems. Math Z 228: 683-703. doi: 10.1007/PL00004636
[6]	Frank RL, König T, Kovařík H (2019) Energy asymptotics in the three-dimensional Brezis-Nirenberg problem. Preprint, arXiv:1908.01331.
[7]	Han ZC (1991) Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann Inst H Poincaré Anal Non Linéaire 8: 159-174.
[8]	Molle R, Pistoia A (2003) Concentration phenomena in elliptic problems with critical and supercritical growth. Adv. Diff. Equations 8: 547-570.
[9]	Rey O (1989) Proof of two conjectures of H. Brezis and L. A. Peletier. Manuscripta Math 65: 19-37. doi: 10.1007/BF01168364
[10]	Rey O (1990) The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent. J Funct Anal 89: 1-52. doi: 10.1016/0022-1236(90)90002-3
[11]	Rodemich E (1966) The Sobolev Inequality with Best Possible Constant, Analysis Seminar Caltech: Spring, 1966.
[12]	Rosen G (1971) Minimum value for c in the Sobolev inequality \|\|φ³\|\| ≤ c\|\|▽φ\|\|³. SIAM J Appl Math 21: 30-32.
[13]	Takahashi F (2004) On the location of blow up points of least energy solutions to the Brezis- Nirenberg equation. Funkc Ekvacioj 47: 145-166. doi: 10.1619/fesi.47.145
[14]	Talenti G (1976) Best constant in Sobolev inequality. Ann Mat Pur Appl 110: 353-372. doi: 10.1007/BF02418013
[15]	Wei J (1998) Asymptotic behavior of least energy solutions to a semilinear Dirichlet problem near the critical exponent. J Math Soc JPN 50: 139-153. doi: 10.2969/jmsj/05010139

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