The principle branch of solutions of a nonlinear elliptic eigenvalue problem on \(\mathbb{R}^ N\)
This paper concerns existence, uniqueness, and asymptotics for the semilinear eigenvalue problem \[ - \Delta u + fu + guh (u) = \lambda u, \quad u > 0 \text{ in } \mathbb{R}^N, \;u (\infty) = 0, \tag{1} \] where \(f \in C^\alpha_{\text{loc}} (\mathbb{R}^N, \mathbb{R})\), \(g \in C^\alpha_{\text{loc}} (\mathbb{R}^N, \mathbb{R}_+) \cap L^\infty (\mathbb{R}^N)\) for some \(\alpha \in (0,1)\), \(h \in C^1 (\mathbb{R}_+, \mathbb{R}_+)\), \(f(x) \to 0\) as \(|x |\to \infty\), \(h\) is strictly increasing, \(h(t) \to 0\), \(th' (t) \to 0\) as \(t \to 0\), and \[ \liminf_{|x |\to \infty} |x |^2 f(x) \geq 0, \quad \sup \bigl\{ - f(x)/g(x) : x \in \mathbb{R}^N \bigr\} < \lim_{t \to \infty} h(t). \] Growth conditions on \(h(t)\) are not required. For \(N = 1,2\), additional hypotheses are imposed on \(g,h\). Let \(\Lambda\) denote the lowest eigenvalue of the corresponding linear problem (i.e., \(g \equiv 0\) in (1)). For \(\Lambda < 0\) and arbitrary \(\lambda \in (\Lambda,0)\), \(a < \sqrt { - \lambda}\), \(p > 1\), the main theorems establish the existence of a unique solution \(u_\lambda \in C^2 (\mathbb{R}^N) \cap W^{2,p} (\mathbb{R}^N)\) of (1) such that \(u_\lambda (x) = o(e^{- a |x |})\) as \(|x |\to \infty\) and the function \(\lambda \to \mu_\lambda\) is continuous from \((\Lambda,0)\) into either \(C^1 (\mathbb{R}^N)\) or \(W^{2,p} (\mathbb{R}^N)\). In addition, an explicit critical value \(p = p(N)\) is found such that the solution branch \(\{(\lambda, u_\lambda) : \lambda \in (\Lambda, 0)\}\) is bounded in \(\mathbb{R} \times W^{2,p} (\mathbb{R}^N)\) for \(p > p(N)\), and unbounded in \(\mathbb{R} \times L^p (\mathbb{R}^N)\) for \(p < p(N)\). The methods include construction of supersolutions for (1).
scientific article; zbMATH DE number 842906
scientific article