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A maximum principle for systems with variational structure and an application to standing waves. (English) Zbl 1331.35124

Let \(\Omega\) be an open connected domain in \(\mathbb{R}^n\) of class \(C^{2,\alpha}\), and let denote by \((s,y)\in \mathbb{R}\times \mathbb{R}^{n-1}\) the generic element of \(\Omega\). Assume that \(\Omega \cap (\{0\}\times \mathbb{R}^{n-1})\) is connected and that there exist \(L>0\) and \(R>0\) such that \((s\pm L,y)\in \Omega\) and \(|y|\leq R\), for each \((s,y)\in \Omega\). Moreover, let \(W:\mathbb{R}^{m}\rightarrow \mathbb{R}\) be a \(C^2\) function satisfying the following conditions:
there exist \(a_-,a_+\in \mathbb{R}^m\), with \(a_-\neq a_+\), such that \(0=W(a_-)=W(a_+)<W(u)\), for all \(u\in \mathbb{R}^m\setminus\{a_-,a_+\}\);
there exist \(r_0>0\) such that the map \(r\in (0,r_0]\rightarrow W(a+r\nu)\) has a strict positive first derivative, for each \(a\in \{a_-,a_+\}\) and for each \(\nu\) in the unit sphere of \(\mathbb{R}^{m-1}\);
there exists \(M>0\) such that \(W(su)\geq W(u)\), for each \(s\in [1,+\infty)\) and \(u\in \{v\in \mathbb{R}^m: |v|=M\}\).
Under the following assumptions, the authors establish the existence of a classical solution \(u:\Omega \rightarrow \mathbb{R}^m\) to the problem
\[ \begin{cases} \Delta u_i = \frac{\partial W}{\partial u_i}(u)&\text{ in }\Omega, \quad i=1,\dots,m,\\\frac{\partial u_i}{\partial n} =0&\text{ on }\partial \Omega, \quad i=1,\dots,m,\\ \lim\limits_{s\rightarrow \pm \infty, (s,y)\in \Omega} u(s,y)=a_{\pm}.\end{cases} \] Moreover, the authors prove that if the quadratic form \(\langle D^2(W(a_+))z,z\rangle\) (resp. \(\langle D^2(W(a_-))z,z\rangle\)) is definite positive, then the solution \(u\) has an exponential decay to \(a_+\) (resp. \(a_-\)), that is there exist \(k_0,K_0>0\) such that
\(|u(s,y)-a_+|\leq K_0e^{-k_0|s|}\) (resp. \(|u(s,y)-a_-|\leq K_0e^{-k_0|s|}\)).
The proof is based on variational methods. The authors first show that the associated energy functional, restricted to the set
\(X_N:=\{u\in W_{\mathrm{loc}}^{1,2}(\Omega;\mathbb{R}^m): |u(s,y)-a_{\pm}|\leq r_0/2\) for \(s\in \mathbb{R}\setminus (-NL,NL) \}\),
where \(N\geq 1\), admits a global minimizer \(u_N\). Then, using a suitable Maximum Principle, which applies to such minimizers, the authors prove that, for \(N\) large enough, the constrains are not saturated, that is one has \(|u(s,y)-a_{\pm}|<r_0/2\). Therefore, for \(N\) large enough, \(u_N\) turns out to be a solution of the problem.

MSC:

35J50 Variational methods for elliptic systems
35J47 Second-order elliptic systems
35J20 Variational methods for second-order elliptic equations
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References:

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