An elliptic system involving p-Laplacian and a chemotactic term

Lucio Boccardo
J. Ignacio Tello
Lourdes Tello

Abstract. We study the nonlinear elliptic system: \[\begin{cases} u \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla u|^{p-2}\nabla u) + u = -\operatorname{div} (a(x) u|\nabla \psi|^{p-2}\nabla \psi) + f(x), \\ \psi \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla \psi |^{p-2}\nabla \psi) = u^{\theta} \end{cases}\] in a bounded, open subset of \(\mathbb{R}^N\) for \(N \gt 2\) and \(2 \lt p \lt N\), where \(f\) satisfies: \[0 \leq f, \quad f\in L^{(p^*){'}}(\Omega), \quad p^*= \frac{Np}{N-p},\] \(a \in L^{\infty}(\Omega) \) is a given function such that there exist \(\alpha, \beta \in \mathbb{R}\) satisfying \[0\lt\alpha \leq a(x) \leq \beta, \quad x\in \Omega.\] We prove the existence of weak solutions in \([W^{1,p}_0(\Omega)]^2\) under the assumption \[0\lt \theta\lt 1- \frac{2}{p^*}.\]

Keywords: nonlinear elliptic systems, weak solutions, \(p\)-Laplacian, chemotaxis term.

Mathematics Subject Classification: 35J47, 35J60, 35D30.

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