Nonexistence of positive radial solutions for a problem with singular potential

Abstract

Abstract. This article completes the picture in the study of positive radial solutions in the function space 𝒟 1 , 2 ( ℝ N ) ∩ L 2 ( ℝ N , | x | - α d x ) ∩ L p ( ℝ N ) ${{\mathcal {D}^{1,2}({\mathbb {R}^N}) \cap L^2({{\mathbb {R}^N}, | x |^{-\alpha } dx})\cap L^p({\mathbb {R}^N})}}$ for the equation - Δ u + A | x | α u = u p - 1 in ℝ N ∖ { 0 } with N ≥ 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.