    March  2009, 8(2): 673-682. doi: 10.3934/cpaa.2009.8.673

## Singular solutions of the Brezis-Nirenberg problem in a ball

 1 Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160 C, Concepción, Chile

Received  December 2007 Revised  June 2008 Published  December 2008

Let $B$ denote the unit ball in $\mathbb R^N$, $N\geq 3$. We consider the classical Brezis-Nirenberg problem

$\Delta u+\lambda u+u^{\frac{N+2}{N-2}} =0 \quad$ in$\quad B$

$u>0 \quad$ in $\quad B$

$u=0 \quad$ on $\quad \partial B$

where $\lambda$ is a constant. It is proven in  that this problem has a classical solution if and only if $\underline \lambda < \lambda < \lambda _1$ where $\underline \lambda = 0$ if $N\ge 4$, $\underline \lambda = \frac{\lambda _1}4$ if $N=3$. This solution is found to be unique in . We prove that there is a number $\lambda_*$ and a continuous function $a(\lambda)\ge 0$ decreasing in $(\underline \lambda, \lambda_*]$, increasing in $[\lambda_*, \lambda_1)$ such that for each $\lambda$ in this range and each $\mu\in (a(\lambda),\infty)$ there exist a $\mu$-periodic function $w_\mu(t)$ and two distinct radial solutions $u_{\mu j}$, $j=1,2$, singular at the origin, with $u_{\mu j}(x)$~$|x|^{-\frac{N-2}2}w_\mu$( log $|x|$) as $x\to 0$. They approach respectively zero and the classical solution as $\mu\to +\infty$. At $\lambda =\lambda_*$ there is in addition to those above a solution ~$c_N|x|^{-\frac{N-2}2}$. This clarifies a previous result by Benguria, Dolbeault and Esteban in , where a existence of a continuum of singular solutions for each $\lambda\in (\underline\lambda, \lambda_1)$ was found.

Citation: Isabel Flores. Singular solutions of the Brezis-Nirenberg problem in a ball. Communications on Pure and Applied Analysis, 2009, 8 (2) : 673-682. doi: 10.3934/cpaa.2009.8.673
  Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943  Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121  Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166  Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41  Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198  Joseph A. Iaia. Localized radial solutions to a semilinear elliptic equation in $\mathbb{R}^n$. Conference Publications, 1998, 1998 (Special) : 314-326. doi: 10.3934/proc.1998.1998.314  Ruofei Yao, Yi Li, Hongbin Chen. Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1585-1594. doi: 10.3934/dcds.2018122  Ying-Chieh Lin, Tsung-Fang Wu. On the semilinear fractional elliptic equations with singular weight functions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2067-2084. doi: 10.3934/dcdsb.2020325  Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091  Paolo Caldiroli. Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$. Communications on Pure and Applied Analysis, 2014, 13 (2) : 811-821. doi: 10.3934/cpaa.2014.13.811  Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155  Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601  Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801  Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439  Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886  David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335  Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080  Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885  Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713  Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447

2020 Impact Factor: 1.916