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Singular solutions of the Brezis-Nirenberg problem in a ball

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  • 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 [3] 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 [17]. 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 [2], where a existence of a continuum of singular solutions for each $\lambda\in (\underline\lambda, \lambda_1)$ was found.

    Mathematics Subject Classification: Primary: 35B33; Secondary: 34C37, 34C20, 35J60.


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