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Super poly-harmonic property of solutions for Navier boundary problems on a half space. (English) Zbl 1288.35230

Summary: In this paper, we consider the following poly-harmonic semi-linear equation with Navier boundary conditions on the half space \(\mathbb R_+^n\): \[ \begin{cases} (-{\Delta})^mu = u^p, \quad p > 1, m \geqslant 1, u > 0, \quad & \text{in } \mathbb{R}_+^n, \\ u = {\Delta}u = \cdots = {\Delta}^{m-1}u = 0, & \text{on } \partial \mathbb{R}_+^n. \end{cases} \tag{1} \] We first prove that the positive solutions of (1) are super poly-harmonic, i.e. \[ (-{\Delta})^iu > 0, \quad i = 0, 1, \dots, m - 1. \tag{2} \] Then, based on (2), we establish the equivalence between PDE (1) and the integral equation \[ u(x) = c_n {\displaystyle \int_{\mathbb{R}_+^n}}\left( \frac{1}{|x - y|^{n-2m}} - \frac{1}{|\bar{x} - y|^{n-2m}}\right)u^p(y)\,dy, \tag{3} \] where \(1 < p < \infty\) and \(\bar{x} = (x_1, \dots, -x_n)\) is the reflection of \(x\) about the boundary. Combining our equivalence result with previous Liouville type theorems on the integral equation (3), we derive the non-existence of positive solutions for problem (1). This in turn enables us to obtain a-priori estimates for the solutions of a family of higher order equations with Navier boundary data on either bounded domains in \(\mathbb{R}^n\) or on Riemannian manifolds with boundaries. A similar super poly-harmonic results like (6) in the whole space \(\mathbb R^n\) has been obtained by J. Wei and X. Xu [Math. Ann. 313, No. 2, 207–228 (1999; Zbl 0940.35082)], however, the method used there can no longer be applied to our situation, hence we introduce some new ideas.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35B45 A priori estimates in context of PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs

Citations:

Zbl 0940.35082
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Full Text: DOI

References:

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