## 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

Zbl 1140.35401
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