A new kind of blowing-up solutions for the Brezis-Nirenberg problem
In this paper, the author is interested in the existence of solutions for the Brezis-Nirenberg problem \[ \begin{cases} -\Delta u = |u|^{p-1}u+\varepsilon\lambda u \quad & \text{ in }\Omega, \\ u=0 & \text{ on }\partial\Omega, \end{cases} \eqno (\mathcal {BN}) \] where \(\varepsilon\) is a small and positive parameter, \(\lambda\in\{-1,1\}, p=\frac{N+2}{N-2}\) and \(\Omega\) is a bounded open domain in \(\mathbb R^N, N>6\), with nontrivial topology and smooth boundary such that the problem \[ \begin{cases} -\Delta u_0 = |u_0|^{p-1}u_0 \quad & \text{ in }\Omega, \\ u_0=0 &\text{ on }\partial\Omega, \\ u_0>0 & \text{ in }\Omega,\end{cases} \eqno(1.1) \] has a positive solution \(u_0\) which is non-degenerate. The author gives an improvement about the existence of positive and sign-changing solutions for \((\mathcal{BN})\) and shows that the result obtained can be extended to the problem \[ \begin{cases} -\Delta u = |u|^{p-1-\lambda\varepsilon} u \quad & \text{ in }\Omega, \\ u=0 & \text{ on } \partial\Omega. \end{cases} \eqno(\mathcal {AC}) \] The following results are obtained. \(\quad\) {Theorem 1.1.} Let \(N>6\) and \(\lambda =1\). (i) Assume that \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\). Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in(0,\varepsilon_0)\), the problem (\(\mathcal{BN}\)) has a sign-changing solution which blows-up negatively at \(\overline{\xi}_0\). (ii) Assume that \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\). Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in(0,\varepsilon_0)\), the problem (\(\mathcal{BN}\)) has a positive solution which blows-up positively at \(\overline{\xi}_0\). (iii) Assume that \(\overline{\xi}_0\in\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\Omega\). Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in(0,\varepsilon_0)\), the problem (\(\mathcal{BN}\)) has a positive solution which blows-up positively at \(\overline{\xi}_0\). {Theorem 1.2.} Let \(N>6\) and \(\lambda =-1\). (i) Assume that \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\). Then there exists \(\varepsilon_0>0\) such that for any \(\varepsilon\in(0,\varepsilon_0)\), the problem (\(\mathcal{BN}\)) has a sign-changing solution with a negative peak at \(\overline{\xi}_0\). All the previous results can be proved also for the almost critical problem (\({\mathcal AC}\)). {Theorem 1.3.} Let \(N>2\) and \(\lambda =1\). (i) If \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\) then the problem (\(\mathcal{AC}\)) has a sign-changing solution with a negative peak at \(\overline{\xi}_0\). ~(ii) If \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\) then the problem (\(\mathcal{AC}\)) has a positive solution with positive peak at \(\overline{\xi}_0\). (iii) If \(\overline{\xi}_0\in\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\Omega\) then the problem (\(\mathcal{AC}\)) has a positive solution with a negative peak at \(\overline{\xi}_0\). {Theorem 1.4.} Let \(N>2\) and \(\lambda =-1\). If \(\overline{\xi}_0\in\partial\Omega\) is a \(C^1-\) stable critical point of the function \(\xi\mapsto \partial_{\nu} u_0(\xi), \xi\in\partial\Omega\) then the the problem (\(\mathcal{AC}\)) has a positive solution with a positive peak at \(\overline{\xi}_0\). The proofs of all the results concerning existence of solutions which blow-up positively or negatively at one point as \(\varepsilon\) goes to zero, rely on Lyapunov-Schmidt reduction scheme. This allows to reduce the problem of finding blowing-up solutions to the problem of finding critical points of a functional which depends only on the blow-up point and the concentration rates. The leading part of the functional is explicitly given in terms of \(u_0\) and of the geometry of the domain, namely the Robin's function.
scientific article; zbMATH DE number 6390176
scientific article