Chapter 1 - Semilinear Elliptic Systems: Existence, Multiplicity, Symmetry of Solutions

Semilinear elliptic systems of the type $-\Delta u=f\left(x,u,v\right),-\Delta v=g\left(x,u,v\right)in\Omega ,$and the more general one, where the nonlinearities depend also on the gradients, namely $-\Delta u=f\left(x,u,v,\nabla u,\nabla v\right),-\Delta v=f\left(x,u,v,\nabla u,\nabla v\right)in\Omega ,$have been object of intensive research recently. In this work we shall discuss some aspects of this research. For the sake of the interested reader we give in Section 10 some references to other topics that are not treated here.

On the above equations u and v are real-valued functions $u,v:\overline{\Omega }\to ℝ$, where Ω is some domain in R^N, N ≥ 3, and Ω¯ its closure. There is also an extensive literature on the case of N = 2, but in the present notes we omit the study of this interesting case; only on Section 4 we make some remarks about this case.

We shall discuss mainly the following questions pertaining to the above systems:

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  Existence of solutions for the Dirichlet problem for the above systems, when Ω is some bounded domain in R^N.

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  Systems with nonlinearities of arbitrary growth.

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  Multiplicity of solutions for problems exhibiting some symmetry.

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  Behavior of solutions at ∞ in the case that Ω is the whole of R^N.

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  Monotonicity and symmetry of positive solutions.

Although we concentrate in the case of the Laplacian differential operator $\Delta ={\displaystyle {\sum }_{j=1}^N\frac{{\partial }^2}{\partial x_j^2}}$, many results stated here can be extended to general second order elliptic operators. Of course, there is the problem of Maximum Principles for systems, which poses interesting questions. See some references on Section 10.

The nonlinearity of the problems appears only in the real-valued functions f, g : Ω¯ × ℝ × ℝ → ℝ. For that matter, problems involving the p-Laplacian are not studied here. Some references are given in section 10.

Here we treat only the Dirichlet problem for the above systems. Other boundary value problems like the Neumann and some nonlinear boundary conditions have been also discussed elsewhere, see Section 10.

Some systems of the type (1.1) can be treated by Variational Methods. In Sections 2 and 3 we study two special classes of such systems, the Gradient systems and the Hamiltonian systems. We say that the system (1.1) above is of the Gradient type if there exists a function F : $\overline{\Omega }$ × ℝ × ℝ → ℝ of class C¹ such that $\frac{\partial F}{\partial u}=f,\frac{\partial F}{\partial v}=g,$and it is said to be of the Hamiltonian type if there exists a function H : $\overline{\Omega }$ × ℝ × ℝ → ℝ of class C¹ such that $\frac{\partial H}{\partial u}=f,\frac{\partial H}{\partial u}=g.$

In Section 2, associated to Gradient systems we have the functional $\Phi \left(u,v\right)=\frac{1}{2}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^2}+\frac{1}{2}{\displaystyle {\int }_{\Omega }{\left|\nabla v\right|}^2}-{\displaystyle {\int }_{\Omega }F\left(x,u,v\right).}$which is defined in the Cartesian product H₀¹ (Ω) × H₀¹ (Ω) provided $F\left(x,u,v\right)\le {\left|u\right|}^p+{\left|v\right|}^q,\forall x\in \Omega ,u,v\in ℝ$with $p,q\le \frac{2N}{N-2}$, if the dimension N ≥ 3.

In Section 3, associated to Hamiltonian systems we will first consider the functional $\Phi \left(u,v\right)={\displaystyle {\int }_{\Omega }\nabla u.\nabla v-{\displaystyle {\int }_{\Omega }H\left(x,u,v\right)}},$which is defined in the Cartesian product H₀¹(Ω) × H₀¹(Ω) provided again that $H\left(x,u,v\right)\le {\left|u\right|}^p+{\left|v\right|}^p,\forall x\in \Omega ,u,v\in ℝ$with $p,q\le \frac{2N}{N-2}$, if the dimension N ≥ 3. However, as we shall see, the restriction on the powers of u and v as above it is too restrictive, in the case of Hamiltonian systems. We shall allow different values of p, q, as observed first in [26] and [84].

In Section 4 we discuss some Hamiltonian systems when one of the nonlinearities may have arbitrary growth following [44].

In Section 5 we present results on the multiplicity of solutions for Hamiltonian systems (1.1) exhibiting some sort of symmetry, we follow [9]. Also in Section 5, following [34], we present a third class of systems which can also be treated by variational methods, namely $\begin{array}{l}-\Delta u=H_u\left(x,u,v\right)in\Omega ,-\Delta v=-H_v\left(x,u,v\right)in\Omega ,\\ u\left(x\right)=v\left(x\right)=0on\partial \Omega .\end{array}$In this form some supercritical systems can be treated, see [34].

In Section 6 we discuss classes of nonvariational systems, which are then treated by Topological Methods. The difficulty here is obtaining a priori bounds for the solutions. There are several methods to tackle this question. We will comment some of them, including the use of Moving Planes and Hardy type inequalities. However the most successful one in our framework seems to be the blow-up method. Here we follow [42]. This method leads naturally to Liouville-type theorems, that is, theorems asserting that certain systems have no nontrivial solution in the whole space R^N or in a half-space R^N₊.

In Section 7, we present some results on Liouville theorems for systems.

In Section 8, systems defined in the whole of R^N are considered again and the behavior of their solutions is presented.

In Section 9 we discuss symmetry and monotonicity of solutions.

And finally in Section 10 we give references to other topics that are not treated here.