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Positive solutions of nonlinear problems involving the square root of the Laplacian. (English) Zbl 1198.35286
The authors study the existence and regularity results of positive solutions for nonlinear problem containing the square root of the Laplacian \(A_{\frac{1}{2}}u=f(u)\) \((A_{\frac{1}{2}}\) stands for the square root of the operator \(-\triangle\) in a bounded domain \(\Omega\)) with zero Dirichlet boundary conditions.
Among other results they prove:
Theorem 1.1 (The existence result). Let \(n\geq 1\) be an integer and \(2^{\sharp}=\frac{2n}{n-1}\) when \(n\geq 2\). Suppose that \(\Omega\) is a smooth bounded domain in \(\mathbb R^n\) and \(f(u)=u^p\). Assume that \(1<p<2^{\sharp}-1=\frac{n+1}{n-1}\) if \(n\geq 2\), or that \(1<p<\infty\) if \(n=1\).
Then, problem admits at least one solution. This solution (as well as every weak solution) belongs to \(C^{2,\alpha}(\overline{\Omega})\) for some \(0<\alpha <1\).
Theorem 1.3 (A priori estimates of Gidas-Spruck type). Let \(n\geq 2\) and \(2^{\sharp}=\frac{2n}{n-1}\). Assume that \(\Omega \subset\mathbb R^n\) is a smooth bounded domain and \(f(u)=u^p\), \(1<p<2^{\sharp}-1=\frac{n+1}{n-1}\).
Then there exists a constant \(C(p,\Omega)\), which depends only on \(p\) and \(\Omega\), such that every weak solution of the problem satisfies \(\|u\|_{L^{\infty}(\Omega)}\leq C(p,\Omega)\).
Theorem 1.6 (Symmetry results of Gidas-Ni-Nirenberg type). Assume that \(\Omega\) is a bounded smooth domain of in \(\mathbb{R}^n\) which is convex in the \(x_1\) direction and symmetric with respect to the hyperplane \(\{x_1=0\}\). Let \(f\) be Lipschitz continuous and \(u\) be a \(C^{2,\alpha}(\overline{\Omega})\) solution of the problem.
Then \(u\) is symmetric with respect to \(x_1\), i.e., \(u(-x_1, x')=u(x_1, x')\) for all \((x_1, x')\in \Omega\). In addition, \(\frac{\partial u}{\partial x_1}<0\) for \(x_1>0\).
In particular, if \(\Omega=B_R(0)\) is a ball, then \(u\) is radially symmetric, \(u=u(|x|)=u(r)\) for \(r=|x|\), and it is decreasing, i.e., \(u_r<0\) for \(0<r<R\).
The introduction contains a detailed review of earlier results and a comparison with the results obtained in the reviewed article.

MSC:
35R11 Fractional partial differential equations
35B45 A priori estimates in context of PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35A25 Other special methods applied to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B09 Positive solutions to PDEs
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