Nodal Blow-Up Solutions to Slightly Subcritical Elliptic Problems with Hardy-Critical Term

Abstract

Abstract The paper is concerned with the slightly subcritical elliptic problem with Hardy-critical term { - Δ ⁢ u - μ ⁢ u | x | 2 = | u | 2 ∗ - 2 - ε ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω $\left\{\begin{aligned} \displaystyle-\Delta u-\mu\frac{u}{|x|^{2}}&% \displaystyle=|u|^{2^{\ast}-2-\varepsilon}u&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega\end{% aligned}\right.$ in a bounded domain Ω ⊂ ℝ N ${\Omega\subset\mathbb{R}^{N}}$ with 0 ∈ Ω ${0\in\Omega}$ , in dimensions N ≥ 7 ${N\geq 7}$ . We investigate the possible blow-up behavior of solutions as μ , ε → 0 ${\mu,\varepsilon\to 0}$ . In particular, we prove the existence of nodal solutions that blow up positively at the origin and negatively at a different point as μ , ε → 0 + ${\mu,\varepsilon\to 0^{+}}$ . The location of the negative blow-up point is determined by the geometry of Ω. Moreover, the asymptotic shape of the solutions is described in detail. An interesting new consequence of our results is that the type of blow-up solutions considered here exists for μ = O ⁢ ( ε α ) ${\mu=O(\varepsilon^{\alpha})}$ with α > N - 4 N - 2 ${\alpha>\frac{N-4}{N-2}}$ . The bound N - 4 N - 2 ${\frac{N-4}{N-2}}$ is sharp.