The ground states of quasilinear Hénon equation with double weighted critical exponents

Abstract

We prove the existence of nontrivial ground state solutions of the critical quasilinear Hénon equation $\displaystyle -\Delta _p u=|x|^{\alpha _1}|u|^{p^{*}(\alpha _1)-2}u-|x|^{\alpha _2}|u|^{p^{*}(\alpha _2)-2}u\ \ {\rm in}\ \mathbb {R}^{N}.$ It is a new problem in the sense that the signs of the coefficients of critical terms are opposite.