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.

Reviewer: Boris V. Loginov (Ul’yanovsk)

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