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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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