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Start Over Subject critical nonlinearity

1. Positive solutions for elliptic problems with critical nonlinearity and combined singularity

Creator:
Chen, Jianqing and Rocha, Eugénio M.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
multiple positive solutions, singular nonlinearity, critical nonlinearity, and Hardy term
Language:
English
Description:
Consider a class of elliptic equation of the form −∆u − λ ⁄ |x| 2 u = u 2 ∗−1 + µu −q in Ω \ {0} with homogeneous Dirichlet boundary conditions, where 0 ∈ Ω ⊂ RN (N ≥ 3), 0 < q < 1, 0 < λ < (N − 2)2 /4 and 2∗ = 2N/(N − 2). We use variational methods to prove that for suitable µ, the problem has at least two positive weak solutions.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Recent progress in attractors for quintic wave equations

Creator:
Savostianov, Anton and Zelik, Sergey
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
damped wave equation, fractional damping, critical nonlinearity, global attractor, and smoothness
Language:
English
Description:
We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of R 3 with damping terms of the form (−∆x) θ ∂tu, where θ = 0 or θ = 1/2. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when θ = 1/2. For θ = 0 existence of smooth attractors is more complicated and follows from Strichartz type estimates.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public