We consider in this paper the nonlinear elliptic equation with Neumann boundary condition
$ \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*} $
For $ a, b\neq 0 $, $ m>\frac{n+1}{n-1} $, $ (n>1) $, $ \eta = \frac{m+1}{2} $ and small data $ f\in L^{\frac{nq}{n+1}, \infty}(\partial \mathbb{R}^{n+1}_{+}) $, $ q = \frac{(n+1)(m-1)}{m+1} $ we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data $ f $ in the function space $ \mathbf{X}^{q}_{\infty} $ where
$ \|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}_{+})}. $
As a direct consequence, we obtain the local regularity property $ C^{1, \nu}_{loc} $, $ \nu\in (0, 1) $ of these solutions as well as energy estimates for certain values of $ m $. Boundary values decaying faster than $ |x|^{-(m+1)/(m-1)} $, $ x\in \mathbb{R}^{n}\setminus\{0\} $ yield solvability and this decay property is shown to be sharp for positive nonlinearities.
Moreover, we are able to show that solutions inherit qualitative features of the boundary data such as positivity, rotational symmetry with respect to the $ (n+1) $-axis, radial monotonicity in the tangential variable and homogeneity. When $ a, b>0 $, the critical exponent $ m_c $ for the existence of positive solutions is identified, $ m_c = (n+1)/(n-1) $.
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