Takahashi, Kazune2022-03-102022-03-102018-11-28Takahashi, K. (2018). Positive solution for Hénon type equations with critical Sobolev growth. <i>Electronic Journal of Differential Equations, 2018</i>(194), pp. 1-17.1072-6691https://hdl.handle.net/10877/15490We investigate the Hénon type equation involving the critical Sobolev exponent with Dirichret boundary condition -∆u = λΨu + |x|α u2*-1 in Ω included in a unit ball, under several conditions. Here, Ψ is a non-trivial given function with 0 ≤ Ψ ≤ 1 which may vanish on ∂Ω. Let λ1 be the first eigenvalue of the Dirichret eigenvalue problem -∆φ = λΨφ in Ω. We show that if the dimension N ≥ 4 and 0 < λ < λ1, there exists a positive solution for small α > 0. Our methods include the mountain pass theorem and the Talenti function.Text17 pages1 file (.pdf)enAttribution 4.0 InternationalCritical Sobolev exponentHenon equationMountain Pass TheoremTalenti functionArticle