Regularity of solutions for an integral system of Wolff type

Abstract

We consider the fully nonlinear integral systems involving Wolff potentials:

(1)$$\{\begin{array}{cc}u(x)=W_{\beta ,\gamma }\left(v^q\right)(x),\hfill & x\in R^n,\hfill \\ v(x)=W_{\beta ,\gamma }\left(u^p\right)(x),\hfill & x\in R^n;\hfill \end{array}$$

where

$$W_{\beta ,\gamma }(f)(x)=\underset{0}{\overset{\infty }{\int }}[\frac{{\int }_{B_t(x)}f(y)dy}{t^{n-\beta \gamma }}{]}^{\frac{1}{\gamma -1}}\frac{dt}{t}.$$

This system includes many known systems as special cases, in particular, when $\beta =\frac{\alpha }{2}$ and $\gamma =2$, system (1) reduces to

(2)$$\{\begin{array}{cc}u(x)=\underset{R^n}{\int }\frac{1}{{|x-y|}^{n-\alpha }}v{(y)}^{q}dy,\hfill & x\in R^n,\hfill \\ v(x)=\underset{R^n}{\int }\frac{1}{{|x-y|}^{n-\alpha }}u{(y)}^{p}dy,\hfill & x\in R^n.\hfill \end{array}$$

The solutions $(u,v)$ of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs

$$\{\begin{array}{cc}{(-\mathrm{\Delta })}^{\alpha /2}u=v^q,\hfill & \text{in}{R}^n,\hfill \\ {(-\mathrm{\Delta })}^{\alpha /2}v=u^p,\hfill & \text{in}{R}^n,\hfill \end{array}$$

which comprises the well-known Lane–Emden system and Yamabe equation.

We obtain integrability and regularity for the positive solutions to systems (1). A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.