Nonexistence of positive radial solutions for a problem with singular potential

Abstract

This article completes the picture in the study of positive radial solutions in the function space 𝒟1,2(ℝN)∩L2(ℝN,|x|-αdx)∩Lp(ℝN)${{\mathcal {D}^{1,2}({\mathbb {R}^N}) \cap L^2({{\mathbb {R}^N}, | x |^{-\alpha } dx})\cap L^p({\mathbb {R}^N})}}$ for the equation -Δu+A|x|αu=up-1inℝN∖{0}withN≥3,A>0,α>0,p>2.$- \Delta u + \frac{A}{| x |^\alpha } u = u^{p-1} \quad \mbox{in } {\mathbb {R}^N}\setminus \lbrace 0\rbrace \mbox{ with } N\ge 3, A> 0, \alpha > 0, p>2. $ An energy balance identity is employed to prove nonexistence of such solutions in the last remaining open region in the (α,p)${{(\alpha , p)}}$ plane.

Keywords

• radial solutions
• singular potential
• nonexistence
• 35j15
• 35j75
• 35j61