Blow-up rates of large solutions for elliptic equations

Abstract

MSC: 35J25 35J65 35J67Keywords: Semilinear elliptic equations Boundary blow-up The first and second expansions of solutions near the boundary The mean curvature of the boundary In this paper, we mainly study the boundary behavior of solutions to boundary blow-up elliptic problems for more general nonlinearities f (which may be rapidly varying at infinitywhere Ω is a bounded domain with smooth boundary in R N , and b ∈ C α (Ω) which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary. Further, when f (s) = s p ± f 1 (s) for s sufficiently large, where p > 1 and f 1 is normalized regularly varying at infinity with index p 1 ∈ (0, p), we show the influence of the geometry of Ω on the boundary behavior for solutions to the problem. We also give the existence and uniqueness of solutions.

Boundary behavior of large solutions to the Monge–Ampère equations with weights

“…Since then many papers have been dedicated to resolving existence, uniqueness and asymptotic behavior issues for solutions to problem (1.3), see, for instance, [1,2,10,11,[14][15][16]27,43,47] and the references therein. Now let us return to problem (1.1).…”

Boundary behavior of large viscosity solutions to infinity Laplace equations

“…Problem (1.4) arises from many branches of mathematics and applied mathematics and has been discussed by many authors and in many contexts, see, for instance, [1,3,7,13,[16][17][18]23,25,[31][32][33][34][35][36]39,42,44,45,49,50] and the references therein. For b ≡ 1 in Ω: when f (u) = e u and N = 2, problem (1.4) was studied much earlier.…”

Asymptotic behavior of boundary blow-up solutions to elliptic equations

“…The problem of second-order approximation of boundary blow-up solutions was first discussed by Lazer and McKenna [30] for a special case f (u) = u p (p > 1) and f (u) = e u when b(x) ≡ 1. This result was developed by many researches, see [1][2][3][4][5][6][7]16,38,48] and the references therein.…”

“…Cîrstea and Rȃdulescu [14], Huang et al [26,28,29], Mi and Liu [37] and Zhang et al [44,45,48] also studied the second-order asymptotic expansion of u to problem (2.9) when (2.10) holds with a suitable combination subclass for k ∈ K and f ∈ RV ρ (ρ > 0). To the best of our knowledge, very few results are known for second-order asymptotic blow-up rates of solutions to (1.1) when f is rapidly varying with index ∞.…”