Boundary Blow-up In Infinity Laplace Equations With T-Varying Nonlinearity

In this paper we study the boundary blow-up solutions to the infinity Laplace equa-tion△∞u=b(x)f(u) in Ω Rn, with the nonlinearity0≤f∈Wloc1,∞[0，∞)Γ-varying at∞, and the weighted function b€C(Ω) positive in Ω and vanishing on the boundary.We quantitatively determine the boundary asymptotic behavior of solutions, relying on the decay rate of b near the boundary and the growth rate of f at infinity. It is shown that the faster b decays to zero, or the more slowly f grows at oo, the faster the solutions go to infinity near the boundary. In addition, we characterize these results via examples possessing various decay rates for b and growth rates for f, and thus obtain a clear picture to illustrate the contribution of the decay of the weighted function b (near the boundary) and the growth of the nonlinearity f (at infinity) to the boundary blow-up rate of the solutions.Chapter1introduces the background and current progress of the field, and briefly describes the contents of the paper. In Chapter2, we give some definitions and auxil-iary results as preliminaries that will be used throughout the paper. In Chapter3, we at first recall the boundary blow-up results for the oc-Laplacian with regularly varying nonlinearity obtained in our another paper, and then state the main results (Theorem1) of the paper for the oo-Laplacian with Γ-varying nonlinearity. Chapter4is devoted to the proof of our main results. Lastly, in Chapter5, we list a series of functions required by the three cases of Theorem1respectively as examples to obtain a clear picture sketching these results.