Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon

Abstract

For a general elliptic problem $-\triangle{u} = g(u)$ in $R^N$ with $N \ge 3$, we show that all solutions have compact support and there exists a least energy solution, which is radially symmetric and decreases with respect to $|x| = r$. With this result we study a singularly perturbed elliptic problem $ -\epsilon^{2} \triangle{u} + |u|^{q-1}u = \lambda u + f(u)$ in a bounded domain $\Omega$ with $0 0 \}$ is a free boundary, where $u_{\epsilon}$ is any nonnegative least energy solution.