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Multiple solutions of semilinear elliptic equations in exterior domains. (English) Zbl 1154.35040

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)].

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators

Citations:

Zbl 1140.35401
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