Exact Morse index of radial solutions for semilinear elliptic equations with critical exponent on annuli

Abstract

Let $$N\ge 3$$ N ≥ 3 , $$R>\rho >0$$ R > ρ > 0 and $$A_{\rho }:=\{x\in \mathbb {R}^N;\ \rho<|x|<R\}$$ A ρ : = { x ∈ R N ; ρ < | x | < R } . Let $$U^{\pm }_{n,\rho }$$ U n , ρ ± , $$n\ge 1$$ n ≥ 1 , be a radial solution with n nodal domains of $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta U+|x|^{\alpha }|U|^{p-1}U=0 &{} \text {in}\ A_{\rho },\\ U=0 &{} \text {on}\ \partial A_{\rho }. \end{array}\right. } \end{aligned}$$ Δ U + | x | α | U | p - 1 U = 0 in A ρ , U = 0 on ∂ A ρ . We show that if $$p=\frac{N+2+2\alpha }{N-2}$$ p = N + 2 + 2 α N - 2 , $$\alpha >-2$$ α > - 2 and $$N\ge 3$$ N ≥ 3 , then $$U^{\pm }_{n,\rho }$$ U n , ρ ± is nondegenerate for small $$\rho >0$$ ρ > 0 and the Morse index $${{\textsf{m}}}(U^{\pm }_{n,\rho })$$ m ( U n , ρ ± ) satisfies $$\begin{aligned} {{\textsf{m}}}(U^{\pm }_{n,\rho }) =n\frac{(N+2\ell -1)(N+\ell -1)!}{(N-1)!\ell !} \quad \text {for small}\ \rho >0, \end{aligned}$$ m ( U n , ρ ± ) = n ( N + 2 ℓ - 1 ) ( N + ℓ - 1 ) ! ( N - 1 ) ! ℓ ! for small ρ > 0 , where $$\ell =[\frac{\alpha }{2}]+1$$ ℓ = [ α 2 ] + 1 . Using Jacobi elliptic functions, we show that if $$(p,\alpha )=(3,N-4)$$ ( p , α ) = ( 3 , N - 4 ) and $$N\ge 3$$ N ≥ 3 , then the Morse index of a positive and negative solutions $${{\textsf{m}}}(U^{\pm }_{1,\rho })$$ m ( U 1 , ρ ± ) is completely determined by the ratio $$\rho /R\in (0,1)$$ ρ / R ∈ ( 0 , 1 ) . Upper and lower bounds for $${{\textsf{m}}}(U^{\pm }_{n,\rho })$$ m ( U n , ρ ± ) , $$n\ge 1$$ n ≥ 1 , are also obtained when $$(p,\alpha )=(3,N-4)$$ ( p , α ) = ( 3 , N - 4 ) and $$N\ge 3$$ N ≥ 3 .