Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, with $N\geq 5$, $a>0$, $\alpha\geq 0$ and $2^*=\frac{2N}{N-2}$. We show that the the exponent $q=\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem $$ \left\{\begin{array}{ll} -\Delta u+au=u^{2^*-1}-\alpha u^{q-1}&\mbox{in}\ \Omega,\\ u>0&\mbox{in}\ \Omega,\\ \frac{\partial u}{\partial\nu}=0&\mbox{on}\ \partial\Omega. \end{array}\right. $$ Namely, we prove that when $q=\frac{2(N-1)}{N-2}$ there exists an $\alpha_{0}>0$ such that the problem has a least energy solution if $\alpha<\alpha_{0}$ and has no least energy solution if $\alpha>\alpha_{0}$.