Decay estimates of positive finite energy solutions to quasilinear and fully nonlinear systems in \(\mathbb{R}^N\) (Q6564433)

The following Wolff type integral system is considered \N\[\N\left\{ \begin{array}{ll} u(x)=R_1(x)W_{\beta,\gamma}(v^p u^r)(x), & u(x)>0,\; x\in\mathbb{R}^{N}, \\\Nv(x)=R_2(x)W_{\beta,\gamma}(u^q v^s)(x), & v(x)>0,\; x\in\mathbb{R}^{N}, \end{array} \right.\tag{1}\N\]\Nwhere \(\gamma>1,\;\beta>0,\; \beta\gamma<N\),\N\[\Np,q>\max\{1,\gamma-1\}, \; r,s\geq 0, \; p-s\geq q-r>-\gamma+1,\N\]\N\[\N\frac{1}{C}\leq R_i(x)\leq C, \;C>0, \; i=1,2.\N\]\NThe Wolff potential of a non-negative Borel measure \(\mu\) on \(\mathbb{R}^{N}\) is defined here by\N\[\NW_{\beta,\gamma}(\mu)(x):=\int\limits_{0}^{\infty}\left[\frac{\mu(B_t(x))}{t^{N-\beta\gamma}}\right]^{\frac{1}{\gamma-1}}\frac{dt}{t},\N\]\Nwhere \(B_t(x)\) is the ball or radius \(t\) centered at point \(x\).\N\NIn the first part, the authors prove the optimal integrability, boundedness and decaying property of finite energy solutions to system (1). In the second part, sharp pointwise estimates of positive finite energy solutions to the \(p\)-Laplacian and \(k\)-Hessian systems related to the above integral system are established. The used methods do not need Harnack type inequalities and they can be applied to deal with solutions without radial structures.