A generalized Pohožaev identity and uniqueness of positive radial solutions ofΔu+g(r)u+h(r)

Uniqueness and nondegeneracy of positive radial solutions of $$\mathbf {div\,{\varvec{(}}{\varvec{\rho }} \nabla u{\varvec{)}} +{\varvec{\rho }}{\varvec{(}}-gu+hu^p{\varvec{)}}=0}$$ div ( ρ ∇ u ) + ρ ( - gu + hu p ) = 0

“…For instance, we consider the problem u − (λ + |x| 2 )u + u p = 0 in R n and u(x) → 0 as |x| → ∞, (1.2) where n ∈ N with n ≥ 2, λ > −n, 1 < p < ∞ in the case n = 2 and 1 < p ≤ (n+2)/(n−2) in the case n ≥ 3. In [47], if n > 2, we could show the uniqueness of positive solutions of (1.2), but we could not show its uniqueness in the case n = 2.…”

“…We can see that (B5) (i) (with other assumptions) in Theorem 1, given in [47], is one of sufficient conditions. We give another sufficient condition (B5) (ii) in Theorem 1.…”

“…These problems include many important equations, like the scalar field equation, and they were studied by many researchers; see [8,9,11,14,[22][23][24][25][26]32,33,[36][37][38]42,44,45,[47][48][49][53][54][55][56] and the references therein. Recently, in [47], we introduced a new generalized Pohožaev function and we studied the uniqueness of positive solutions of problem (1.1).…”

“…Recently, in [47], we introduced a new generalized Pohožaev function and we studied the uniqueness of positive solutions of problem (1.1). We showed the result is applicable to various examples.…”

“…This paper is organized as follows. In the next section, we recall the generalized Pohožaev identity introduced in [47]. In Sect.…”