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Semilinear elliptic systems. (English) Zbl 0955.35020
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).
From the introduction: In this series of lectures we survey and discuss results on the existence of solutions for the system \[ -\Delta u= g(x,u,v),\quad -\Delta v= f(x,u,v)\quad\text{in }\Omega, \] subject to Dirichlet boundary conditions on \(\partial\Omega\). \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 3\). Due to the use of some results on the regularity of solutions of elliptic problems we shall, at some point, assume implicitly some regularity on \(\partial\Omega\). We do not discuss the case \(N= 2\), where the imbedding theorems of Trudinger-Moser allow the treatment of nonlinearities which have a growth faster than the polynomial growth required by the Sobolev imbeddings.
Clearly we do not present in full the proofs of all the results discussed. The methodology used here is the following. We state the results, comment the key points in the proofs, explaining the techniques used, compare the results and hint questions that can be object of further study. With these purposes in mind, we do not state the more general results available in the literature. For that matter, some results are valid for more general second-order operators instead of the Laplacian. Also other boundary conditions can be considered. A careful guide to the literature is presented all along these lectures.
We emphasize that we will be mostly interested in superlinear systems. For instance if \(f\) and \(g\) have growth with respect to \(u\) and \(v\) faster than linear. For Hamiltonian systems we will have a notion of superlinearity which takes into account that we have a coupled system. It is expected that superlinear systems, as the case of superlinear equations, are much harder to study, no matter which method one uses. Variationally, we are confronted with questions of compactness of the functional, as well as with an intrincated geometry, in most cases. Topologically, the question of a priori bounds comes as a difficult problem. We address all these questions.
In Lecture 1, we discuss a class of non-variational systems, showing how to prove the existence of positive solutions. We start there the discussion of a priori bounds, which is taken in detail in Lecture 2, using the blow-up method. This discussion leads naturally to questions which are answered by theorems of Liouville type. And such theorems are the object of Lecture 3. In Lecture 4 we discuss gradient systems and in Lecture 5 we consider Hamiltonian systems.
For the entire collection see [Zbl 0941.00024].

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
47J25 Iterative procedures involving nonlinear operators
35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A15 Variational methods applied to PDEs
35B45 A priori estimates in context of PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations