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Existence of a solution to a coupled elliptic system. (English) Zbl 0791.35043

Suppose \(\Omega\) is a bounded regular domain of \(\mathbb{R}^ N\), \(N>1\), \(\partial\Omega\) its boundary, \(\sigma\), \(\lambda\) and \(f\) are bounded functions from \(\Omega\times\mathbb{R}\) to \(\mathbb{R}\), continuous with respect to \(y\in\mathbb{R}\) for a.e. \(x\in\Omega\), and measurable with respect to \(x\in\Omega\) for any \(y\in\mathbb{R}\), and such that: \[ \exists\alpha>0;\quad \alpha\leq \sigma(\cdot,y) \text{ and } \alpha\leq \lambda(\cdot,y), \qquad \forall y\in\mathbb{R}, \quad \text{for a.e. } x\in\Omega. \] The aim of this work is to prove the following existence theorem: Theorem. Under the above assumptions, there exists a solution \((u,\varphi)\) satisfying: \[ u\in \textstyle {\bigcap_{p<{N\over {N-1}}}} W_ 0^{1,p}(\Omega), \qquad \varphi\in H_ 0^ 1(\Omega) \]
\[ \begin{aligned} \int_ \Omega &\sigma (x,u(x)) D\varphi(x) D\psi(x)dx= \int_ \Omega f(x,u(x))\psi(x)dx,\qquad \forall\psi\in H_ 0^ 1 (\Omega),\\ \int_ \Omega &\lambda(x,u(x)) Du(x) Dv(x)dx= \int_ \Omega \sigma(x,u(x)) D\varphi(x) D\varphi(x)v(x)dx, \quad \forall v\in \bigcup_{q>N} W_ 0^{1,q}(\Omega).\end{aligned} \] {}.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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