We discuss the existence of a maximizer for a maximizing problem associated with the Trudinger-Moser type inequality in R N (N ≥ 2). Different from the bounded domain case, we obtain both of the existence and the nonexistence results. The proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences. M. Ishiwata (B)
Let 1 ≤ p ≤ ∞. We show that a function u ∈ C(R N ) is a viscosity solution to the normalized p-Laplace equation n p u(x) = 0 if and only if the asymptotic formulaholds as ε → 0 in the viscosity sense. Here,with respect to λ ∈ R. This kind of asymptotic mean value property (AMVP) extends to the case p = 1 previous (AMVP)'s obtained when μ p (ε, u)(x) is replaced by other kinds of mean values. The natural definition of μ p (ε, u)(x) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of u. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic p-Laplace equation.
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