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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
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