Existence of solutions of semilinear elliptic problems on unbounded domains
This paper is devoted to the study of the existence of positive solutions \(u\in H^1_0(\Omega)\) of the semilinear elliptic equation \[ - \Delta u+ \lambda u= |u|^{p- 2} u\quad\text{in }\Omega,\tag{\(*\)} \] where \(\Omega\) is an unbounded domain in \(\mathbb{R}^N\), \(\lambda\in \mathbb{R}\), \(N\geq 2\) and \(p\in (2, 2^*)\), where \(2^*= 2N/(N- 2)\) for \(N\geq 3\) and \(2^*= \infty\) for \(N= 2\). Let \[ f(u)= {1\over 2} \int_\Omega (|\nabla u|^2+ \lambda u^2)dx,\;\alpha(\Omega):= \inf\Biggl\{f(u): u\in H^1_0(\Omega), \int_\Omega |u|^p dx= 1\Biggr\}. \] Section 2 provides various tools for the study of this problem, like the following result: if \(\Omega_1\subset \Omega_2\), then \(\alpha(\Omega_2)\leq \alpha(\Omega_1)\), and if \(\Omega_1\subsetneqq a_1\) and \(\alpha(\Omega_1)\) admits a minimizer, i.e. there exists a solution \(u\in H^1_0(\Omega)\) of \((*)\) (with \(\Omega= \Omega_1\)) such that \(f(u)= \alpha(\Omega_1)\) and \(\int_\Omega |u|^p dx= 1\), then \(\alpha(\Omega_2)< \alpha(\Omega_1)\). The authors prove that \(\alpha(\Omega)= \alpha\) if \(\Omega\subsetneqq \mathbb{R}^N\) is a ball up domain (Sections 2,3) i.e. if for any \(r> 0\) there exists \(x\in \Omega\) such that \(B(x; r)\subset \Omega\). Using an adaptation of a compactness result by P. L. Lions, they also prove that if \(\Omega\) is a periodic domain (Section 4) which means that there exists a partition \(\{Q_m\}\) of \(\Omega\) and points \(\{y_m\}\) in \(\mathbb{R}^N\) such that \(\{y_m\}\) forms a subgroup of \(\mathbb{R}^N\), \(Q_0\) is bounded and \(Q_m= Q_0 y_m\), then \(\alpha(\Omega)\) admits a minimizer and therefore \(\alpha(\Omega)> \alpha(\mathbb{R}^N)\). This situation covers several cases already known in the literature and extends them. Section 5 is devoted to finite perturbed periodic domains, a case which offers various interesting examples and is especially interesting in view of the non-existence results by Esteban-Lions. The main tool of this section is the following result: let \(\Omega_1, \Omega_2\subset \mathbb{R}^N\) be domains such that \(\Omega_1\cap \Omega_2\) is bounded. If \(\alpha(\Omega_1)\leq \alpha(\Omega_2)\) and if \(\alpha(\Omega_1)\) admits a minimizer, then \(\alpha(\Omega_1\cup \Omega_2)\) admits a minimizer. The last section is devoted to exterior spherical flask domains \(\{(x_1, x')\in \mathbb{R}\times \mathbb{R}^{N- 1}: R^2< x^2_1|x'|^2, |x'|< r\}\) and refers to a paper by \textit{C. C. Chan} and \textit{H. C. Wang} [to appear in Nonlinear Anal. Theory, Methods, Appl.]. This paper provides a systematic approach for various situations involving unbounded domains. It covers a lot of partial results and extends them.
scientific article; zbMATH DE number 459209