A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry

Abstract

In this paper we consider the following elliptic equation: \begin{equation*} \begin{cases} \; -\Delta u + u = f(x,u) \qquad \text{in}\; \mathbf{R}^N, \\ \quad u \in H^1(\mathbf{R}^N). \end{cases} \end{equation*} Here $f(x,u)$ is invariant under a finite group action $G \subset O(N)$ which acts on $S^{N-1}$ effectively. When $N \geq 3$, we show the existence of a positive solution without global assumptions like: $ u \mapsto \frac{f(x,u)}{u} $ is increasing in $ u > 0$, $f(x,u) \geq f^{\infty}(u)$. We can deal with asymptotically linear equations as well as superlinear equations. Interaction estimates between solution at infinity play important roles in our argument.