## Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems.(English)Zbl 1165.35360

From the introduction: This paper is a contribution to the study of boundary value problems for systems of elliptic partial differential equations of the form
$\begin{cases} -\Delta u_1=f(x,u_1,u_2) &\text{in }\Omega,\\ -\Delta u_2= g(x,u_1,u_2) &\text{in }\Omega,\\ u_1=u_2=0 &\text{on }\partial\Omega, \end{cases} \tag{1}$ where $$u_1$$, $$u_2$$ are real-valued functions defined on a smooth bounded domain $$\Omega$$ in $$\mathbb R^N$$, $$N\geq 3$$, and $$f$$ and $$g$$ are Hölder continuous functions defined in $$\overline{\Omega}\times\mathbb R\times\mathbb R$$.
This type of systems has been extensively studied during the last two decades – see for example the survey paper [D. G. de Figueiredo, in: Ambrosetti, A. (ed.) et al., Proceedings of the 2nd school on nonlinear functional analysis and applications to differential equations, ICTP, Trieste, Italy, April 21-May 9, 1997. Singapore: World Scientific. 122–152 (1998; Zbl 0955.35020)] and the references therein. One of the important questions is the existence of a priori bounds for positive smooth solutions of these systems.
It is well known that the existence of a priori bounds depends on the growth of the functions $$f$$ and $$g$$ as $$u_1$$ and $$u_2$$ go to infinity. In view of what is known for scalar equations, one expects that some polynomial (subcritical) growth is to be required. In fact such a restriction comes from the Sobolev imbedding theorems in dimension $$N\geq 3$$. It is also known that a priori bounds are particularly interesting when superlinear equations are considered.
The simplest case of systems of type (1) – which is the only case in which a priori bounds have been studied up to now – is when the leading parts of $$f$$ and $$g$$ involve just pure powers of $$u_1$$ and $$u_2$$. More precisely, when $$f$$ and $$g$$ are such that (1) can be written in the form
$\begin{cases} -\Delta u_1= a(x)u_1^{\alpha_{11}}+ b(x)u_2^{\alpha_{12}}+ h_1(x,u_1,u_2),\\ -\Delta u_2= c(x)u_1^{\alpha_{21}}+ d(x)u_2^{\alpha_{22}}+ h_2(x,u_1,u_2), \end{cases}\tag{2}$
where the exponents $$\alpha_{ij}$$ are nonnegative real numbers, $$a(x)$$, $$b(x)$$, $$c(x)$$, $$d(x)$$ are nonnegative continuous functions on $$\overline{\Omega}$$, and $$h_1,h_2$$ are locally bounded functions such that uniformly in $$x\in\Omega$$
$\begin{cases} \displaystyle \lim_{|(u_1,u_2)|\to\infty} \big(a(x)u_1^{\alpha_{11}}+ b(x)u_2^{\alpha_{12}}\big)^{-1} |h_1(x,u_1,u_2)|=0,\\ \displaystyle \lim_{|(u_1,u_2)|\to\infty} \big(c(x)u_1^{\alpha_{21}}+ d(x)u_2^{\alpha_{22}}\big)^{-1} |h_2(x,u_1,u_2)|=0. \end{cases} \tag{3}$
The method used here in order to obtain the a priori bounds, the so-called blow-up method, was introduced in [B. Gidas and J. Spruck, Commun. Partial Differ. Equations 6, 883–901 (1981; Zbl 0462.35041)] to treat the scalar case. Let us note that the blow-up method itself depends on results of nonexistence of positive solutions of equations and systems in the whole space or in a half-space. Such results are usually referred to as Liouville type theorems – see Section 2.
Our main result unifies and extends the previous results on a priori bounds for (2). In addition, it allows more general nonlinearities in systems of type (1), namely mixed powers of $$u_1$$ and $$u_2$$ in the principal part of the nonlinearities $$f$$ and $$g$$.

### MSC:

 35J57 Boundary value problems for second-order elliptic systems 35B45 A priori estimates in context of PDEs 35J60 Nonlinear elliptic equations

### Citations:

Zbl 0955.35020; Zbl 0462.35041
Full Text:

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