Asymptotics for semilinear field equations with slowly decaying mass (Q1323067)

The objective of this note is to describe the asymptotic behaviour at infinity of positive energy solutions \(u(x)\) of semilinear elliptic equations, including the type \[ -\Delta u+m(x)u = f(x)u^ \alpha,\quad x \in \Omega \] in an exterior domain \(\Omega \subset \mathbb{R}^ N\), \(N \geq 3\), \(1<\alpha \leq (N+2)/(N-2)\). It is assumed that \(m\) and \(f\) are nonnegative, bounded locally Hölder continuous functions in \(\Omega\). If \(m(x) \sim m_ 0 | x |^{-\sigma}\) as \(| x | \to \infty\) for some constants \(m_ 0>0\) and \(\sigma \in (0,2)\), the authors show in particular that every positive finite energy solution \(u(x)\) decays faster than \(C | x|^{-\gamma}\) but slower than \(C \exp(- \mu | x |)\) for any positive constants \(C,\mu\) and \(\nu\). Sharper results are given under refined hypotheses.