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February  2022, 21(2): 517-539. doi: 10.3934/cpaa.2021186

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

 Rheinische Friedrich-Wilhelms-Universität Bonn, Endeniche Allee 60, 53115, Bonn, Germany

Received  July 2021 Revised  September 2021 Published  February 2022 Early access  November 2021

Fund Project: The author is supported by the DAAD through the program "Graduate School Scholarship Programme, 2018" (Number 57395813) and by the Hausdorff Center for Mathematics at Bonn

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,
 $m_c = (n+1)/(n-1)$
.
Citation: Gael Diebou Yomgne. On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space. Communications on Pure and Applied Analysis, 2022, 21 (2) : 517-539. doi: 10.3934/cpaa.2021186
##### References:
 [1] D. R. Adam, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa, 25 (1971), 203-217. [2] D. H. Armitage, The Neumann problem for a function harmonic in $\mathbb{R}^{n}\times (0, \infty)$, Arch. Rational Mech. Anal., 63 (1976), 89-105.  doi: 10.1007/BF00280145. [3] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1976. [4] M. Comte and M. C. Knaap, Solutions of elliptic equations involving critical Sobolev exponents with neumann boundary conditions, Manuscripta math., 69 (1990), 43-70.  doi: 10.1007/BF02567912. [5] E. Constantin and N. H. Pavel, Green function of the Laplacian for the Neumann problem in $\mathbb{R}^{n}_{+}$, Libert. Math., 30 (2010), 57-69. [6] J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232.  doi: 10.1016/j.jfa.2010.12.028. [7] M. F. de Almeida and L. S. M. Lima, Adams' trace principle on Morrey-type spaces over $\beta$-Hausdorff dimensional surfaces, 46 (2021), 1161–1177. [8] M. F. de Almeida and L. S. M. Lima, Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces, arXiv: 1807.04122. [9] G. Diebou Yomgne, Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials, Nonlinear Anal., 215 (2022), 112623. [10] J. F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate, Commun. Pure Appl. Math., 43 (1990), 857-883.  doi: 10.1002/cpa.3160430703. [11] J. F. Escobar, Yamabe problem on manifolds with boundary, J. Differ. Geom., 35 (1992), 21-84. [12] E. B. Fabes, M. Jr. M. Jodeit and M. N. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math., 141 (1978), 165-186.  doi: 10.1007/BF02545747. [13] L. C. F. Ferreira, E. S. Medeiros and M. Montenegro, On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space, Calc. Var. Partial Differ. Equ., 47 (2013), 667-682.  doi: 10.1007/s00526-012-0531-2. [14] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. d'Analyse Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551. [15] Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Differ. Integral Equ., 12 (1999), 601-612. [16] E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Institute Math., 234 (2001), 1-383. [17] R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142. [18] P. Quittner and W. Reichel, Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions, Calc. Var. Partial Differ. Equ., 32 (2008), 429-452.  doi: 10.1007/s00526-007-0155-0. [19] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Basel, Birkhäuser, Amsterdam, 2007. [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [21] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse, Vol. 34, Springer-Verlag, Berlin Heidelberg, 1996. doi: 10.1007/978-3-662-03212-1. [22] S. Tang, L. Wang and M. Zhu, Nonlinear elliptic equations on the upper half space, arXiv: 1906.03739. [23] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Annali Scuola Norm. Sup. Pisa, 22 (1968), 265-274.

show all references

##### References:
 [1] D. R. Adam, Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa, 25 (1971), 203-217. [2] D. H. Armitage, The Neumann problem for a function harmonic in $\mathbb{R}^{n}\times (0, \infty)$, Arch. Rational Mech. Anal., 63 (1976), 89-105.  doi: 10.1007/BF00280145. [3] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1976. [4] M. Comte and M. C. Knaap, Solutions of elliptic equations involving critical Sobolev exponents with neumann boundary conditions, Manuscripta math., 69 (1990), 43-70.  doi: 10.1007/BF02567912. [5] E. Constantin and N. H. Pavel, Green function of the Laplacian for the Neumann problem in $\mathbb{R}^{n}_{+}$, Libert. Math., 30 (2010), 57-69. [6] J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232.  doi: 10.1016/j.jfa.2010.12.028. [7] M. F. de Almeida and L. S. M. Lima, Adams' trace principle on Morrey-type spaces over $\beta$-Hausdorff dimensional surfaces, 46 (2021), 1161–1177. [8] M. F. de Almeida and L. S. M. Lima, Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces, arXiv: 1807.04122. [9] G. Diebou Yomgne, Nonlinear biharmonic equation in half-space with rough Neumann boundary data and potentials, Nonlinear Anal., 215 (2022), 112623. [10] J. F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities and an eigenvalue estimate, Commun. Pure Appl. Math., 43 (1990), 857-883.  doi: 10.1002/cpa.3160430703. [11] J. F. Escobar, Yamabe problem on manifolds with boundary, J. Differ. Geom., 35 (1992), 21-84. [12] E. B. Fabes, M. Jr. M. Jodeit and M. N. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math., 141 (1978), 165-186.  doi: 10.1007/BF02545747. [13] L. C. F. Ferreira, E. S. Medeiros and M. Montenegro, On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space, Calc. Var. Partial Differ. Equ., 47 (2013), 667-682.  doi: 10.1007/s00526-012-0531-2. [14] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. d'Analyse Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551. [15] Y. Lou and M. Zhu, Classification of nonnegative solutions to some elliptic problems, Differ. Integral Equ., 12 (1999), 601-612. [16] E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Institute Math., 234 (2001), 1-383. [17] R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142. [18] P. Quittner and W. Reichel, Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions, Calc. Var. Partial Differ. Equ., 32 (2008), 429-452.  doi: 10.1007/s00526-007-0155-0. [19] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Basel, Birkhäuser, Amsterdam, 2007. [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [21] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse, Vol. 34, Springer-Verlag, Berlin Heidelberg, 1996. doi: 10.1007/978-3-662-03212-1. [22] S. Tang, L. Wang and M. Zhu, Nonlinear elliptic equations on the upper half space, arXiv: 1906.03739. [23] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Annali Scuola Norm. Sup. Pisa, 22 (1968), 265-274.
The region below the critical curve $M_c(n)$ (resp. $m_c(n)$ below $M_c(n)$) indicates the nonexistence range relative to Eq. (1.2) (with $f = 0$ and $a, b\neq 0$ having same sign) (resp. for Eq. (1.2) with $a, b>0$ and $f\neq 0$)
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