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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