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 possible energy of a sign-changing solution to the nonlinear scalar field equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="0.17em" /> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>D</m:mi> <m:mn>1,2</m:mn> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi mathvariant="normal">R</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> -Δu=f(u), u∈ D1,2(RN), where N ≥ 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 12 c 0 if N = 5, 6 and smaller than 10 c 0 if N ≥ 7, where c 0 is the ground state energy.