Decay estimates for Wolff potentials in R-N and gradient-dependent quasilinear elliptic equations

作者：Car Siegfried^*

来源：Journal of Differential Equations, 2018, 265(8): 3685-3708.

DOI：10.1016/j.jde.2018.05.015

We prove decay estimates of positive weak solutions u is an element of D-1,D- p(R-N) boolean AND C-1(R-N) and their gradients vertical bar del u vertical bar of the p-Laplacian problem (1 < p < N):
u is an element of D-1,D- p(R-N): -Delta(p)u = a(x),
where the measurable function a satisfies the decay estimate (alpha > 0):
0< a(x) less than or similar to 1/1+vertical bar x vertical bar(N+alpha),for all x is an element of R-N.
The positive Borel measure on R-N generated by the right-hand side function a and its associated Wolff potential is known to be the main tool for pointwise estimates of u(x) > 0 and vertical bar del u(x)vertical bar. Therefore, one of the main goal of this paper is to prove decay estimates for corresponding Wolff potentials in R-N. The obtained decay estimates in conjunction with a sub-supersolution method, which is developed here for quasilinear elliptic equations, are then applied to prove the existence of positive solutions for classes of gradient-dependent quasilinear equations in the form
u is an element of D-1,D- p(R-N) : - Delta(p)u = a(x) f(u, del u).

• 出版日期2018-10-15