A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS

作者：Cabre Xavier^*; Sanchon Manel; Spruck Joel

来源：Discrete and Continuous Dynamical Systems, 2016, 36(2): 601-609.

DOI：10.3934/dcds.2016.36.601

• Semi-stable solutions
• extremal solutions
• regularity
• boundedness
• semilinear elliptic equations

We consider positive semistable solutions u of Lu + f(u) = 0 with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is an element of C-2 is a positive, nondecreasing, and convex nonlinearity which is super-linear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension n <= 9, but only established for n <= 4. In this paper we prove the L-infinity bound up to dimension n = 5 under the following further assumption on f: for every epsilon > 0, there exist T = T(epsilon) and C = C(epsilon) such that f '(t) < C f(t)(1+epsilon) for all t > T. This bound will follow from a L-p-estimate for f ' (u) for every p < 3 (and for all n >= 2). Under a similar but more restrictive assumption on f, we also prove the L-infinity estimate when n = 6. We remark that our results do not assume any lower bound on f '.

• 出版日期2016-2