Exact multiplicity for the perturbed Q-curvature problem in $${\mathbb{R}^N, N \geq 5}$$ R N , N ≥ 5

Abstract

Let N ≥ 5 and $${{\mathcal{D}}^{2,2} (\mathbb{R}^N)}$$ denote the closure of $${C_0^\infty (\mathbb{R}^N)}$$ in the norm $${\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}$$ Let $${K \in C^2 (\mathbb{R}^N).}$$ We consider the following problem for ɛ ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P ɛ ) for all small ɛ > 0.