Singular Limit Of P-Laplace Equation

This paper is concerned with the singular limit of the minimal solutions, as p→1,λ→∞, of the quasilinear Neumann problem involving p-Laplace operatorwhereΩis a bounded smooth domain in Rⁿ, n≥2, 1 < p < n, 1 < q 0 is a parameter.Define for (?)∈W^1.p(Ω),(?)≠0,In particular, when p = 1,where∫_Ω|D(?)|dx is the total variation of (?) inΩ.Because the equation has many physical applications , the case where 1 < p < n has been extensively studied recently, as well as the singular limit of its minimal solutions. However, when p = 1 andλ→∞, little is known. We consider this problem in the first part of this paper and investigate the asymptotic behavior of the minimizer of Q_1.λ. The main result is the following.Theorem 1 Assume n≥2,1 < q < n/n-1, if u_λis the minimizer of Q_1.λ and‖u_λ‖(?)(Ω)=1, then asλ→∞,we have:(i) u_λ→0 in L¹(Ω), so there is a subsequence u_λ→0 a.e.,(ii)∫_Ω|Du_λ|dx→+∞,it follows Q_1.λ→+∞,(iii)‖u_λ‖L^∞(Ω)→+∞ On the other hand, the case 1 < p < n is studied extensively, while when p > n, little is known. So we we consider the special situation as n =1,p > 1 and find some new phenomena which is not known as n≥2. At the same time, we get a Holder estimate of the minimal solution when p near 1, which is different from n≥2. The main results are the following.Theorem 2 Assume n > 1, q > 1,I = [0,1], u_p is the minimal solution when p > 1. x₀ isan arbitrary point in I. Then we have(i)For 0 < R < min{x₀, 1 -x₀}, there exist 0 <Ï„< 1, A > 0 depending only onλ, R, suchfor1 < p_j≤3/2, 0 < r < R, it holds that(ii)(?) < p≤3/2,(?)(a,b)(?)(?)I, for 0 <Ï„< 1, there existsα(p) =Ï„/p+ 2-2/p andC = C(a, b) > 0,which is independent of p, such thatTheorem3 Assume n > 1,p > 1,q > 1,u_p is the minimal solution when p > 1, and u_λis the singular limit of u_p as p→1. If u_p satisfiesThen there exists M > 0, such that...