We consider the following elliptic problems on simply connected planar domains: u u −Δ >= u 0 0 = |x|2αK(x)up in Ω u −Δ = u 0 = λ|x|2αK(x)eu in on Ω ∂Ω; in Ω on ∂Ω with α > −1, λ > 0, p > 1, 0 < K(x) ∈ C1 (Ω). We show that any solution to each problem must satisfy a uniform bound on the mass, which is given, respectively, by λ∫Ω |x|2αK(x)eudx and p ∫Ω |x|2αK(x)up+1dx. The same results apply to some systems and more general nonlinearities. The proofs are based on the Riemann mapping theorem and a Pohožaevtype identity.
Battaglia, L. (2019). Uniform bounds for solutions to elliptic problems on simply connected planar domains. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 147(10), 4289-4299 [10.1090/proc/14482].
Uniform bounds for solutions to elliptic problems on simply connected planar domains
BATTAGLIA, Luca
2019-01-01
Abstract
We consider the following elliptic problems on simply connected planar domains: u u −Δ >= u 0 0 = |x|2αK(x)up in Ω u −Δ = u 0 = λ|x|2αK(x)eu in on Ω ∂Ω; in Ω on ∂Ω with α > −1, λ > 0, p > 1, 0 < K(x) ∈ C1 (Ω). We show that any solution to each problem must satisfy a uniform bound on the mass, which is given, respectively, by λ∫Ω |x|2αK(x)eudx and p ∫Ω |x|2αK(x)up+1dx. The same results apply to some systems and more general nonlinearities. The proofs are based on the Riemann mapping theorem and a Pohožaevtype identity.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
