Abstract
In this paper we study the positive solutions of the equation −Δu + λu = f(u) in a bounded symmetric domain Ω in ℝN, with the boundary condition u = 0 on ∂Ω. Using the maximum principle we prove the symmetry of the solutions v of the linearized problem. From this we deduce several properties of v and u; in particular we show that if N = 2 there cannot exist two solutions which have the same maximum if f is also convex and that there exists only one solution if f(u) = up and λ = 0.In the final section we consider the problem −Δu = uP + μuq in Ω with u = 0 on ∂Ω, and show that if 1 < p <N+2N-2, q ϵ]0,1[ there are exactly two positive solutions for μ sufficiently small and some particular domain Ω.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
