Nonexistence of radial node solutions for elliptic problems with critical Sobolev exponents

Abstract

Let Ω ⊂ R n be a ball centered at the origin with radius 1 and 3 ≤ n ≤ 6 , 2 ∗ = 2 n n − 2 and μ ∈ [ 0 , ( n − 2 2 ) 2 ) . By using a well-known Pohozaev-type identity and some ODE techniques, we obtained the nonexistence of sign-changing radial solutions for elliptic problems with critical Sobolev and Hardy terms − Δ u = μ | x | 2 u + | u | 2 ∗ − 2 u + λ u on Ω , u ∈ H 0 1 ( Ω ) for suitable small positive number λ . Meanwhile, the nonexistence of radial sign-changing solutions for p -Laplace problems with critical Sobolev exponent is also obtained.