On the Location of Blow up Points of Least Energy Solutions to the Brezis-Nirenberg Equation

Takahashi, Futoshi

doi:10.1619/fesi.47.145

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…Mainly, our proof is almost the same as in [9] in which we treated the case when k ≡ 1, and the argument there originates from [5]. But there is also some improvement compared to the former calculations in [9].…”

supporting

confidence: 53%

“…Concentration phenomena in elliptic problems involving critical Sobolev exponents like (P ε,k ) are now widely studied. For the special case of k ≡ 1, see [4], [7], [8] and [9]. The Robin function of the domain plays an important role in these studies.…”

mentioning

confidence: 99%

“…Later in [9], it is proved that any blow up point of least energy solutions is a minimum point of the Robin function on general bounded domains in R N , N ≥ 4.…”

mentioning

confidence: 99%

“…Furthermore, we need the appropriate bound of the value S ε n ,k from the above. The following Lemma is proved by the same argument of Lemma 2.7 in [9], so we omit the proof.…”

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Concentration points of least energy solutions to the Brezis-Nirenberg equation with variable coefficients

Takahashi

2006

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confidence: 53%

mentioning

confidence: 99%

“…Later in [9], it is proved that any blow up point of least energy solutions is a minimum point of the Robin function on general bounded domains in R N , N ≥ 4.…”

mentioning

confidence: 99%

“…Furthermore, we need the appropriate bound of the value S ε n ,k from the above. The following Lemma is proved by the same argument of Lemma 2.7 in [9], so we omit the proof.…”

mentioning

confidence: 99%

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Concentration points of least energy solutions to the Brezis-Nirenberg equation with variable coefficients

Takahashi

2006

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“…Asymptotics (1.7) and (1.8) in the case where V is a negative constant are essentially contained in [30]; see also [32] for related results. The case of general V ∈ C( ) can be treated by similar methods.…”

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confidence: 98%

Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

2021

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For a bounded open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ S ( a + ϵ V ) = inf 0 ≢ u ∈ H 0 1 ( Ω ) ∫ Ω ( | ∇ u | 2 + ( a + ϵ V ) | u | 2 ) d x ( ∫ Ω u 6 d x ) 1 / 3 involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of $$S(a+\epsilon V)-S$$ S ( a + ϵ V ) - S as $$\epsilon \rightarrow 0+$$ ϵ → 0 + , where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $$S(a+\epsilon V)<S$$ S ( a + ϵ V ) < S for all sufficiently small $$\epsilon >0$$ ϵ > 0 .

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Asymptotic behavior of large solutions to H-systems with perturbations

Takahashi¹

2004

Nonlinear Analysis: Theory, Methods & Applications

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On the Location of Blow up Points of Least Energy Solutions to the Brezis-Nirenberg Equation

Cited by 7 publications

References 9 publications

Concentration points of least energy solutions to the Brezis-Nirenberg equation with variable coefficients

Concentration points of least energy solutions to the Brezis-Nirenberg equation with variable coefficients

Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

Asymptotic behavior of large solutions to H-systems with perturbations

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