Ngô Quốc Anh

June 11, 2010

The method of moving planes: Elliptic systems in the whole space

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 5:13

Li Ma and Baiyu Liu recently announced a new result accepted in Advances in Mathematics journal [here]. In that elegant paper, they considered the following system of equations $\displaystyle\left\{ \begin{gathered} - \Delta u = g(u,v), \hfill \\ - \Delta v = f(u,v), \hfill \\ u > 0,v > 0, \hfill \\ \end{gathered} \right.$

in the whole space $\mathbb R^n$ with $n\geqslant 3$ where $f$ and $g$ are two smooth functions in $\mathbb R^2_+$.

This work is closely related to works done in [here] and [here]. Precisely, in [here], the authors have proved that classical solution $(u, v)$ of the PDEs is symmetric about some points $x_1, x_2 \in \mathbb R^n$ respectively, under assumptions

1. $u(x) \to 0$, $v(x) \to 0$ as $|x| \to \infty$;

2. $\displaystyle\frac{{\partial g}}{{\partial v}}(u,v) \geqslant 0, \quad \frac{{\partial f}}{{\partial u}} (u,v)\geqslant 0,\quad\forall (u,v) \in \left[ {0, + \infty } \right) \times \left[ {0, + \infty } \right)$;

3. $\displaystyle\frac{{\partial g}}{{\partial u}}(0,0) < 0, \quad \frac{{\partial f}}{{\partial v}}(0,0) < 0$;

4. $\displaystyle\det \left( {\begin{array}{*{20}{c}} {\frac{{\partial g}}{{\partial u}}} & {\frac{{\partial g}}{{\partial v}}} \\ {\frac{{\partial f}}{{\partial u}}} & {\frac{{\partial f}}{{\partial v}}} \\ \end{array} } \right)(0,0) > 0$.

In this paper, the authors relax the hypothesis (4), for the price of supposing exact growth of the solutions at infinity, and of the nonlinearities at zero. Precisely, they proved

Theorem. Let $(u, v)$ be a classical solution of the system and $f,g \in C^1([0,\infty) \times [0,\infty), \mathbb R)$. Suppose

1. $\displaystyle u(x) \sim \frac{1}{{{{\left| x \right|}^\alpha }}}, \quad v(x) \sim \frac{1}{{{{\left| x \right|}^\beta }}}, \quad \left| x \right| \to \infty$;
2. $\displaystyle \frac{{\partial g}}{{\partial u}}\lesssim {u^{p - 1}},\quad\frac{{\partial f}}{{\partial v}} \lesssim {v^{q - 1}},\quad u \to {0^ + },\quad v \to {0^ + }$;
3. $\displaystyle \frac{{\partial g}}{{\partial v}}\lesssim {u^{a - 1}},\quad\frac{{\partial f}}{{\partial u}} \lesssim {v^{b - 1}},\quad u \to {0^ + },\quad v \to {0^ + }$;
4. $\displaystyle \frac{{\partial g}}{{\partial v}} > 0,\quad\frac{{\partial f}}{{\partial u}} > 0,\quad\forall (u,v) \in (0,\infty ) \times (0,\infty )$;

where positive constants $\alpha, \beta, p, q, a, b$ satisfy $\displaystyle\alpha (p - 1) > 2,\beta (q - 1) > 2,\alpha (a - 1) > 2,\beta (b - 1) > 2$.

Then there exists a point in $x_0 \in \mathbb R^n$, such that $u(x)=u(|x-x_0|), \quad v(x)=v(|x-x_0|)$.

The proof used is the method of moving planes. This kind of system is also a generalization of the Lane-Emden system given by $\displaystyle\left\{ \begin{gathered} - \Delta u = {v^b }, \hfill \\ - \Delta v = {u^a }. \hfill \\ \end{gathered} \right.$

It can also be applied to the following system $\displaystyle\left\{ \begin{gathered} - \Delta u = - {u^p} + {v^b}, \hfill \\ - \Delta v = {u^a} - {v^q}. \hfill \\ \end{gathered} \right.$

However, it is no longer able to apply to $\displaystyle\left\{ \begin{gathered} - \Delta u = {u^p} - {v^b}, \hfill \\ - \Delta v = - {u^a} + {v^q}. \hfill \\ \end{gathered} \right.$

In fact, hypothesis (4) plays an important role in their argument. To our knowledge, the latter case is still open.