Decay estimates of positive finite energy solutions to quasilinear and fully nonlinear systems in RN

In this paper, we first investigate the regularity of finite energy solutions to the following Wolff type integral system: {u(x) = R-1 (x)W-beta,W-gamma(v(p)u(r))(x), u(x), > 0 , x is an element of R-N, v(x) = R-2 (x)W-beta,W-gamma(u(q)v(s))(x), v(x) > 0, x is an element of R-N, where gamma > 1, beta> 0, beta gamma < N , W-beta,W-gamma is the Wolff potential, R-1, R-2 are double bounded functions in R-N and p, q > max {1, gamma- 1}, r, s >= 0 with p- s >= q- r >- gamma + 1. We exploit the regularity lifting lemma to obtain the optimal integrability, boundedness and decaying property of finite energy solutions to the system. Secondly, we establish sharp pointwise estimates of positive finite energy solutions to the p-Laplacian and k-Hessian systems related to the above integral system, by using the previous regularity results, the interior Holder estimates of solutions for the corresponding differential systems and a doubling lemma of Polacik, Quittner and Souplet (Duke Math. J.,2007). These extend Vetois's decay results (Indiana Univ. Math. J.,2019) on the positive solutions of Laplacian systems to the p-Laplacian and k-Hessian cases. We remark that our methods do not need Harnack type inequalities and it can be applied to deal with the solutions without radial structures. As far as we know, this is the first attempt to derive sharp decay estimates for possibly non-radial solutions involving k-Hessian operators. We also obtain some decay estimates of the gradients at infinity. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.