Lin, Huei-Li Multiple solutions of semilinear elliptic equations in exterior domains. (English) Zbl 1154.35040 Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 3, 531-549 (2008). Let \(\Omega\) denote a smooth exterior domain in \(\mathbb{R}^N\), \(q:\Omega\to\mathbb{R}\) be positive and continuous, and \(p\in(2,2^*)\), where \(2^*=\infty\) if \(N=1,2\) and \(2^*=2N/(N-2)\) if \(N\geq3\). The author considers solutions to the problem \[ \begin{cases} -\Delta u+u = q(x)| u| ^{p-2}u,&\text{in }\Omega,\\ u\in H^1_0(\Omega). \end{cases}\tag{1} \]It is assumed that \(q_\infty:=\lim_{| x| \to\infty}q(x)>0\) exists and that \(q\) is not constant. The following results are proved:Theorem 1: Suppose that there are \(C>0\) and \(\delta\in(0,2)\) such that \(q(x)\geq q_\infty+C e^{-\delta| x| }\). Suppose moreover that \(q\) is bounded. Then (1) has two positive solutions.Theorem 2: Suppose that there are \(C>0\) and \(\delta\in(0,1)\) such that \(q(x)\geq q_\infty+C e^{-\delta| x| }\). Then (1) has two solutions, one positive and one sign changing.One should compare these theorems with the complementary result in M. Clapp and T. Weth [Commun. Partial Differ. Equations 29, No. 9–10, 1533–1554 (2004; Zbl 1140.35401)]. Reviewer: Nils Ackermann (México) Cited in 4 Documents MSC: 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 47J30 Variational methods involving nonlinear operators Keywords:exterior domain; superlinear subcritical problem; positive solution; sign changing solution Citations:Zbl 1140.35401 × Cite Format Result Cite Review PDF Full Text: DOI