On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space

Abstract

<p style='text-indent:20px;'>We consider in this paper the nonlinear elliptic equation with Neumann boundary condition</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \begin{cases} \Delta u = a|u|^{m-1}u\, \, \mbox{ in }\, \, \mathbb{R}^{n+1}_{+}\\ \dfrac{\partial u}{\partial t} = b|u|^{\eta-1}u+f\, \, \mbox{ on }\, \, \partial \mathbb{R}^{n+1}_{+}. \end{cases} \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M1">\begin{document}$ a, b\neq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ m>\frac{n+1}{n-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ (n>1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \eta = \frac{m+1}{2} $\end{document}</tex-math></inline-formula> and small data <inline-formula><tex-math id="M5">\begin{document}$ f\in L^{\frac{nq}{n+1}, \infty}(\partial \mathbb{R}^{n+1}_{+}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ q = \frac{(n+1)(m-1)}{m+1} $\end{document}</tex-math></inline-formula> we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data <inline-formula><tex-math id="M7">\begin{document}$ f $\end{document}</tex-math></inline-formula> in the function space <inline-formula><tex-math id="M8">\begin{document}$ \mathbf{X}^{q}_{\infty} $\end{document}</tex-math></inline-formula> where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \|u\|_{ \mathbf{X}^{q}_{\infty}} = \sup\limits_{t>0}t^{\frac{n+1}{q}-1}\|u(t)\|_{L^{\infty}( \mathbb{R}^{n})}+\|u\|_{L^{\frac{q(m+1)}{2}, \infty}( \mathbb{R}^{n+1}_{+})}+\|\nabla u\|_{L^{q, \infty}( \mathbb{R}^{n+1}_{+})}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>As a direct consequence, we obtain the local regularity property <inline-formula><tex-math id="M9">\begin{document}$ C^{1, \nu}_{loc} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \nu\in (0, 1) $\end{document}</tex-math></inline-formula> of these solutions as well as energy estimates for certain values of <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula>. Boundary values decaying faster than <inline-formula><tex-math id="M12">\begin{document}$ |x|^{-(m+1)/(m-1)} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ x\in \mathbb{R}^{n}\setminus\{0\} $\end{document}</tex-math></inline-formula> yield solvability and this decay property is shown to be sharp for positive nonlinearities.</p><p style='text-indent:20px;'>Moreover, we are able to show that solutions inherit qualitative features of the boundary data such as positivity, rotational symmetry with respect to the <inline-formula><tex-math id="M14">\begin{document}$ (n+1) $\end{document}</tex-math></inline-formula>-axis, radial monotonicity in the tangential variable and homogeneity. When <inline-formula><tex-math id="M15">\begin{document}$ a, b>0 $\end{document}</tex-math></inline-formula>, the critical exponent <inline-formula><tex-math id="M16">\begin{document}$ m_c $\end{document}</tex-math></inline-formula> for the existence of positive solutions is identified, <inline-formula><tex-math id="M17">\begin{document}$ m_c = (n+1)/(n-1) $\end{document}</tex-math></inline-formula>.</p>