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Sign-changing tower of bubbles for the Brezis-Nirenberg problem. (English) Zbl 1336.35170
In this paper, the authors study the following Brezis-Nirenberg problem $\begin{cases} -\Delta u =|u|^{p-1}u+\varepsilon u &\text{ in }\Omega,\\ \quad\,\, u = 0 &\text{ on } \partial \Omega,\end{cases}\tag{1}$ where $$\Omega$$ is a bounded smooth domain of $$\mathbb R^N$$ with $$N\geq 7$$, $$\varepsilon$$ supposed to be small and positive and $$p+1=\frac{2N}{N-2}$$ is the critical Sobolev exponent for the embedding of $$H^1_0(\Omega)$$ into $$L^{p+1}(\Omega).$$
The authors construct a sign-changing solution of (1) when the domain $$\Omega$$ is symmetric with respect to $$N$$ orthogonal axis.

##### MSC:
 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 35J20 Variational methods for second-order elliptic equations
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