caa a22 4500 605496137 CHVBK 20210128100534.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10231-014-0438-y doi (NATIONALLICENCE)springer-10.1007/s10231-014-0438-y Iacopetti Alessandro Dipartimento di Matematica e Fisica, Universitá degli Studi di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Rome, Italy aut Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem [Elektronische Daten] [Alessandro Iacopetti] 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. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Semilinear elliptic equations nationallicence Critical exponent nationallicence Sign-changing radial solutions nationallicence Asymptotic behavior nationallicence Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/6(2015-12-01), 1649-1682 0373-3114 194:6<1649 2015 194 10231 https://doi.org/10.1007/s10231-014-0438-y text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0438-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Iacopetti Alessandro Dipartimento di Matematica e Fisica, Universitá degli Studi di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Rome, Italy aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/6(2015-12-01), 1649-1682 0373-3114 194:6<1649 2015 194 10231