An upper bound for the least energy of a sign-changing solution to a zero mass problem

We give an upper bound for the least energy of a sign-changing solution to the the nonlinear scalar field equation $$-\mathrm{\Delta }u=f(u),u\in D^{1,2}({\mathbb{R}}^N),$$ where $N\ge 5$ and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that $12c_0$ if $N=5,6$ and smaller than $10c_0$ if $N\ge 7$, where $c_0$ is the ground state energy.