The present paper addresses existence of a ground state solution for the equation \[ X^q_N\Delta u+ u^{p-1}= 0,\quad u(x)> 0,\quad x\in\Omega,\tag{1} \] where \(\Omega\subset Y:= \mathbb R^{N-1}\times (0,\infty)\), \(N\geq 2\) and \(2^*= {2N\over N-2}\) for \(N> 2\), \(2^*= \infty\) for \(N= 2\). More precisely, the author shows that (1) admits a ground state solution of (1) for \(q= N-{p(N-2)\over 2}\), \(p\in (2,2^*)\).