Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem
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| 024 | 7 | 0 | |a 10.1007/s10231-014-0438-y |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10231-014-0438-y | ||
| 100 | 1 | |a Iacopetti |D Alessandro |u Dipartimento di Matematica e Fisica, Universitá degli Studi di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Rome, Italy |4 aut | |
| 245 | 1 | 0 | |a Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem |h [Elektronische Daten] |c [Alessandro Iacopetti] |
| 520 | 3 | |a We study the asymptotic behavior, as $$\lambda \rightarrow 0$$ λ → 0 , of least energy radial sign-changing solutions $$u_\lambda $$ u λ , of the Brezis-Nirenberg problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u + |u|^{2^* -2}u &{}\quad \hbox {in}\ B_1\\ u=0 &{}\quad \hbox {on}\ \partial B_1, \end{array}\right. \end{aligned}$$ - Δ u = λ u + | u | 2 ∗ - 2 u in B 1 u = 0 on ∂ B 1 , where $$\lambda >0,\, 2^*=\frac{2n}{n-2}$$ λ > 0 , 2 ∗ = 2 n n - 2 and $$B_1$$ B 1 is the unit ball of $$\mathbb {R}^n,\, n\ge 7$$ R n , n ≥ 7 . We prove that both the positive and negative part $$u_\lambda ^+$$ u λ + and $$u_\lambda ^-$$ u λ - concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover, we show that suitable rescalings of $$u_\lambda ^+$$ u λ + and $$u_\lambda ^-$$ u λ - converge to the unique positive regular solution of the critical exponent problem in $$\mathbb {R}^n$$ R n . Precise estimates of the blow-up rate of $$\Vert u_\lambda ^\pm \Vert _{\infty }$$ ‖ u λ ± ‖ ∞ are given, as well as asymptotic relations between $$\Vert u_\lambda ^\pm \Vert _{\infty }$$ ‖ u λ ± ‖ ∞ and the nodal radius $$r_\lambda $$ r λ . Finally, we prove that, up to constant, $$\lambda ^{-\frac{n-2}{2n-8}} u_\lambda $$ λ - n - 2 2 n - 8 u λ converges in $$C_{\mathrm{loc}}^1(B_1-\{0\})$$ C loc 1 ( B 1 - { 0 } ) to $$G(x,0)$$ G ( x , 0 ) , where $$G(x,y)$$ G ( x , y ) is the Green function of the Laplacian in the unit ball. | |
| 540 | |a Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 | ||
| 690 | 7 | |a Semilinear elliptic equations |2 nationallicence | |
| 690 | 7 | |a Critical exponent |2 nationallicence | |
| 690 | 7 | |a Sign-changing radial solutions |2 nationallicence | |
| 690 | 7 | |a Asymptotic behavior |2 nationallicence | |
| 773 | 0 | |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/6(2015-12-01), 1649-1682 |x 0373-3114 |q 194:6<1649 |1 2015 |2 194 |o 10231 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10231-014-0438-y |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10231-014-0438-y |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Iacopetti |D Alessandro |u Dipartimento di Matematica e Fisica, Universitá degli Studi di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Rome, Italy |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/6(2015-12-01), 1649-1682 |x 0373-3114 |q 194:6<1649 |1 2015 |2 194 |o 10231 | ||