A Study On The Asymptotic Properties Of Solutions To Quasilinear Elliptic Boundary Value Problems

In this paper, we study the asymptotic properties of solutions to the fol-lowing quasilinear elliptic problems by Karamata regular variation theory and sub-supersolution method. where Ω is a bounded domain with smooth boundary in RN, λ≥ 0, b, a E Clocα(Ω) (α∈(0,1)), are positive in Ω and may be vanishing or singular on the boundary. The function g ∈C1((0,∞,(0,∞)) satisfies limtâ†'0+g(t)= ∞, and f∈C([0,∞),[0,∞)); where Ω, g and the weight b, a both satisfy the hypotheses in (P1), λ ∈ R, q ∈[0,2] and σ ∈ Clocα(Ω) (0< α< 1); where Ω, still satisfies the hypothesis in (P1), the weight b ∈ C(Ω) is a nonnegative and nontrivial function, and may be vanishing or singular on the boundary, the nonlinearity f ∈ C([0,∞)) is a positive nondecreasing function on (0,∞).When b, a and g, f satisfy some proper hypotheses, we study the second order expansion of the classical solution near the boundary to problem (P1) and reveal that a(x)f(u) does not affect the expansion. This implies that every classical solution has the same second order behavior near the boundary. On the other hand, we introduced a more general kind of Karamata regular variation functions Λμ,β,μ≥0β∈R which expend the results of Zhijun Zhang [1] and Ling Mi and Bin Liu [2]. Especially, when μ> 0 and β= 0, we improve the precision of the second expansion in [2].Next, we consider the existence and asymptotic behavior of the classical solution to problem (P2).These conclusions expend the results of Zeddini et al. [3], Maagli and Zribi [4], Maagli [5], Zhijun Zhang [7], Zhijun Zhang et al. [8], Bo Li and Zhijun Zhang [9], Dupaigne, Ghergu and Radulescu [10].Finally, we study two aspects to problem (P3) as below:(I) We study the first order expansion of the weak solutions to problem (P3) near the boundary. These results highly expend the results of Mohammed [11], Zhijun Zhang et al. [12], Zhijun Zhang and Ling Mi [13]. Significantly,f ∈RVp-1 is the critical condition for the existence of the weak solution to (P3), i.e., if Keller-Osserman condition in general is true, then there exist weak solutions to (P3). Otherwise, there exists no weak solution to (P3).(II) When Df ∈ (0,1/p), we study the influence of the mean curvature of the boundary to the second order expansion of boundary blow-up solutions to problem (P3), which expend the results of Shuibo Huang and Qiaoyu Tian [14], Shuibo Huang et al. [15], Zhijun Zhang [16], Ling Mi and Bin Liu [17] and Repovs [18]. The non-linear property of operator (â–³p) (p> 1) is not the main resistance for our study. But, to the second expansion, it is very difficult to deal with it. In 2012, Repovs gave the solution to this resistance under the condition that the mean curvature of (?)Ω is a constant. In this paper, we will give out the entire solution when the mean curvature of (?)Ω is a general function.