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 $\mathrm{\Omega }$ be a smooth bounded domain in ${\mathbb{R}}^N$, with $N\ge 5$, $a>0$, $\alpha \ge 0$ and $2^{\ast }=\frac{2N}{N-2}$. 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 $$\{\begin{array}{ll}-\mathrm{\Delta }u+au=u^{2^{\ast }-1}-\alpha u^{q-1}& in\mathrm{\Omega },\\ u>0& in\mathrm{\Omega },\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0& on\mathrm{\partial }\mathrm{\Omega }\mathrm{.}\end{array}$$ 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 $\alpha >{\alpha }_0$.