Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

• Published in: Mathematics in Engineering
• Main Authors: Clapp, Mónica, Pacella, Filomena
• Format: Article in Journal/Newspaper
• Language: English
• Published: 2021
• Subjects: Critical exponent; Neumann boundary condition; nonradial solutions; North Pole
• Online Access: http://hdl.handle.net/11573/1464120; https://doi.org/10.3934/mine.2021022

We study a critical Neumann problem in an unbounded cone Σ_ω:={tx:x∈ω and t>0}, where ω is an open connected subset of the unit sphere S^N−1 in R^N with smooth boundary, N≥3 and 2∗:=2N/N−2. We assume that some local convexity condition at the boundary of the cone is satisfied. If ω is symmetric with respect to the north pole of S^N−1, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone Σ_ω∩B_1(0) is large enough (but possibly smaller than half the volume of the unit ball B_1(0) in R^N), we establish the existence of a positive nonradial solution.