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The maximum principles for fractional Laplacian equations and their applications. (English) Zbl 1373.35330

Summary: This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: \[ \begin{cases}(-\Delta)^{\frac{\alpha}{2}}u=f(x,u,u) & \text{ in }\Omega,\\ u>0,\text{ in }\Omega;\quad u\equiv 0,& \quad\text{ in }\mathbb R^n\backslash\Omega\end{cases}\quad\text{ for }\alpha\in (0,2). \] Here \(\Omega\) is a domain (bounded or unbounded) in \(\mathbb R^n\) which is convex in \(x_1\)-direction. \((-\Delta)^{\frac{\alpha}{2}}\) is the nonlocal fractional Laplacian operator which is defined as \[ (-\Delta)^{\frac{\alpha}{2}}u(x)=C_{n,\alpha}\text{P.V.} \int_{\mathbb R^n}\frac{u(x)-u(y)}{|x-y|^{n+\alpha}},\quad 0<\alpha<2. \] Under various conditions on \(f(x,u,\mathbf p)\) and on a solution \(u(x)\) it is shown that \(u\) is strictly increasing in \(x_1\) in the left half of \(\Omega\), or in \(\Omega\). Symmetry (in \(x_1\)) of some solutions is proved.

MSC:

35R11 Fractional partial differential equations
35B50 Maximum principles in context of PDEs
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