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Qualitative properties of positive solutions to systems of quasilinear elliptic equations. (English) Zbl 1325.35066

The authors study monotonicity properties of positive solutions to some quasilinear elliptic systems involving \(p\)-Laplace equations in half spaces. To be more precise, let \(u,v \in C^{1,\alpha} \left( \overline{\mathbb{R}^N_+}\right)\) be solutions to the problem \[ \begin{cases} -\Delta_pu+a|\nabla u|^r =f(u,v) \quad & \text{in }\mathbb{R}^N_+,\\ -\Delta_qv+b|\nabla v|^s=g(u,v) \quad & \text{in }\mathbb{R}^N_+,\\ u(x',y)>0,\quad v(x',y)>0\quad & \text{in }\mathbb{R}^N_+,\\ u(x',0)=0,\quad v(x',0)=0\quad & \text{on }\partial \mathbb{R}^N_+, \end{cases}\tag{1} \] where \(N\geq 2\), \(1<p,q<2\), \(1<r\leq p\), \(1<s\leq q\), \(a,b \geq 0\) and the maps \(f,g: \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) satisfy the following assumptions:
(H1)
\(f\) and \(g\) are locally Lipschitz functions on \(\mathbb{R} \times \mathbb{R}\);
(H2)
\(f(z,t), g(z,t)>0\) for all \(z,t>0\);
(H3)
\(f(z,0)=f(0,t)=g(z,0)=g(0,t)=0\) for all \(z,t\in \mathbb{R}\);
(H4)
\(\frac{\partial f}{\partial t}(z,t)\geq 0\) and \(\frac{\partial g}{\partial z}(z,t)\geq 0\) for a.a.\(z,t\in \mathbb{R}\) if \(a=b=0\). If \(a \neq 0\) or \(b\neq 0\), it is assumed that \(f\) is strictly increasing in the second variable and \(g\) is strictly increasing in the first variable.
Denoting a point in \(\mathbb{R}^N_+\) by \(x=(x',y)\) with \(x'=(x_1,x_2, \dots, x_{N-1})\) and \(y=x_N\), the main result in this paper reads as follows.
Theorem: Let hypotheses (H1)–(H4) be satisfied and let \(u,v \in C^{1,\alpha}_{\text{loc}}\left(\overline{\mathbb{R}^N_+} \right)\) be positive solutions of problem (1) with \(1<p,q<2\). Further assume that \(|\nabla u|, |\nabla v| \in L^\infty\left(\mathbb{R}^N_+ \right)\). Then, \(u\) and \(v\) are monotone increasing with respect to the \(y\)-direction, that is \[ \frac{\partial u}{\partial y} \geq 0, \quad \frac{\partial v}{\partial y} \geq 0 \quad \text{in } \mathbb{R}^N_+. \]

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B09 Positive solutions to PDEs
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Full Text: Euclid