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

Abstract

Let $\Omega$ 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 $$ \left\{\begin{array}{ll} -Δu+au=u^{2^*-1}-αu^{q-1}&\mbox{in}\ Ω,\\ u>0&\mbox{in}\ Ω,\\ \frac{\partial u}{\partialν}=0&\mbox{on}\ \partialΩ. \end{array}\right. $$ 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 $\alpha >{\alpha }_0$.

30 pages

References in corpus (1)

• A sharp inequality for Sobolev functions

Cited by in corpus (1)

• A sharp inequality for Sobolev functions