Asymptotic boundary estimates to infinity Laplace equations with Γ-varying nonlinearity

Abstract

In a previous paper of the authors (Wang et al. (2014) [40]), the asymptotic estimates of boundary blow-up solutions were established to the infinity Laplace equation Δ∞u=b(x)f(u) in Ω⊂RN, with the nonlinearity 0≤f∈C[0,∞) regularly varying at ∞, and the weighted function b∈C(Ω¯) positive in Ω and vanishing on the boundary. The present paper gives a further investigation on the asymptotic behavior of boundary blow-up solutions to the same equation by assuming f to be Γ-varying. Note that a Γ-varying function grows faster than any regularly varying function. We first quantitatively determine the boundary blow-up estimates with the first expansion, relying on the decay rate of b near the boundary and the growth rate of f at infinity, and further characterize these results via examples possessing various decay rates for b and growth rates for f. In particular, we pay attention to the second-order estimates of boundary blow-up solutions. It was observed in our previous work that the second expansion of solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the classical Laplacian. The second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow-up solutions between the infinity Laplacian and the classical Laplacian.