We are concerned with the problem with Minkowski-curvature operator on an exterior domain
$ \begin{align} \left\{\begin{array}{ll} -\text{div}\Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) = \lambda K(|x|)\frac{f(u)}{u^\gamma}\ \ \ &\text{in}\ B^c,\\[2ex] \frac{\partial u}{\partial n}|_{\partial B^c} = 0, \ \ \lim\limits_{|x|\to\infty}u(x) = 0, \end{array} \right. \end{align} \quad\quad\quad\quad\quad (P) $
where $ 0\leq\gamma < 1 $, $ B^c = \{x\in \mathbb{R}^N: |x| > R\} $ is a exterior domain in $ \mathbb{R}^N $, $ N > 2 $, $ R > 0 $, $ K\in C([R, \infty), (0, \infty)) $ is such that $ \int_R^\infty rK(r)dr < \infty $, the function $ f:[0, \infty)\to (0, \infty) $ is a continuous function such that $ \lim\limits_{s\to\infty}\frac{f(s)}{s^{\gamma+1}} = 0 $ and $ \lambda > 0 $ is a parameter. We show that problem $ (P) $ has at least one positive radial solution for all $ \lambda > 0 $. The proof of our main result is based upon the method of sub and super solutions.
Citation: Zhongzi Zhao, Meng Yan. Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain[J]. AIMS Mathematics, 2023, 8(9): 20654-20664. doi: 10.3934/math.20231052
We are concerned with the problem with Minkowski-curvature operator on an exterior domain
$ \begin{align} \left\{\begin{array}{ll} -\text{div}\Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) = \lambda K(|x|)\frac{f(u)}{u^\gamma}\ \ \ &\text{in}\ B^c,\\[2ex] \frac{\partial u}{\partial n}|_{\partial B^c} = 0, \ \ \lim\limits_{|x|\to\infty}u(x) = 0, \end{array} \right. \end{align} \quad\quad\quad\quad\quad (P) $
where $ 0\leq\gamma < 1 $, $ B^c = \{x\in \mathbb{R}^N: |x| > R\} $ is a exterior domain in $ \mathbb{R}^N $, $ N > 2 $, $ R > 0 $, $ K\in C([R, \infty), (0, \infty)) $ is such that $ \int_R^\infty rK(r)dr < \infty $, the function $ f:[0, \infty)\to (0, \infty) $ is a continuous function such that $ \lim\limits_{s\to\infty}\frac{f(s)}{s^{\gamma+1}} = 0 $ and $ \lambda > 0 $ is a parameter. We show that problem $ (P) $ has at least one positive radial solution for all $ \lambda > 0 $. The proof of our main result is based upon the method of sub and super solutions.
| [1] |
R. Bartnik, L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys., 87 (1982), 131–152. https://doi.org/10.1007/bf01211061 doi: 10.1007/bf01211061
|
| [2] |
C. Bereanu, P. Jebelean, J. Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc., 137 (2009), 161–169. https://doi.org/10.1090/s0002-9939-08-09612-3 doi: 10.1090/s0002-9939-08-09612-3
|
| [3] |
C. Bereanu, P. Jebelean, J. Mawhin, Multiple solutions for Neumann and periodic problems with singular $\phi$-Laplacian, J. Funct. Anal., 261 (2011), 3226–3246. https://doi.org/10.1016/j.jfa.2011.07.027 doi: 10.1016/j.jfa.2011.07.027
|
| [4] |
C. Bereanu, P. Jebelean, P. J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644–659. https://doi.org/10.1016/j.jfa.2013.04.006 doi: 10.1016/j.jfa.2013.04.006
|
| [5] |
G. Dai, J. Yao, F. Li, Spectrum and bifurcation for semilinear elliptic problems in ${\mathbb{R}}^n$, J. Differ. Equations, 263 (2017), 5939–5967. https://doi.org/10.1016/j.jde.2017.07.004 doi: 10.1016/j.jde.2017.07.004
|
| [6] |
C. Gerhardt, H-surfaces in Lorentzian manifolds, Commun. Math. Phys., 89 (1983), 523–553. https://doi.org/10.1007/BF01214742 doi: 10.1007/BF01214742
|
| [7] |
J. A. Iaia, Existence of infinitely many solutions for semilinear problems on exterior domains, Commun. Pure Appl. Anal., 19 (2020), 4269–4284. https://doi.org/10.3934/cpaa.2020193 doi: 10.3934/cpaa.2020193
|
| [8] |
J. A. Iaia, Existence and nonexistence of sign-changing solutions for singular semilinear equations on exterior domains, Nonlinear Anal., 217 (2022), 112752. https://doi.org/10.1016/j.na.2021.112752 doi: 10.1016/j.na.2021.112752
|
| [9] |
E. Ko, E. Lee, R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain, Discrete Contin. Dyn. Syst., 33 (2013), 5153–5166. https://doi.org/10.3934/dcds.2013.33.5153 doi: 10.3934/dcds.2013.33.5153
|
| [10] |
R. Ma, H. Gao, Y. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, J. Funct. Anal., 270 (2016), 2430–2455. https://doi.org/10.1016/j.jfa.2016.01.020 doi: 10.1016/j.jfa.2016.01.020
|
| [11] |
R. Ma, Y. Zhu, Y. Zhang, L. Yang, Spectrum and global bifurcation results for nonlinear second-order problem on all of $\mathbb{R}$, Ann. Funct. Anal., 14 (2023), 3. https://doi.org/10.1007/s43034-022-00226-0 doi: 10.1007/s43034-022-00226-0
|
| [12] |
F. Obersnel, P. Omari, S. Rivetti, Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal. Real World Appl., 13 (2012), 2830–2852. https://doi.org/10.1016/j.nonrwa.2012.04.012 doi: 10.1016/j.nonrwa.2012.04.012
|
| [13] |
M. Pei, L. Wang, Positive radial solutions of a mean curvature equation in Lorentz-Minkowski space with strong singularity, Appl. Anal., 99 (2020), 1631–1637. https://doi.org/10.1080/00036811.2018.1555322 doi: 10.1080/00036811.2018.1555322
|
| [14] | W. Rudin, Real and complex analysis, 3 Eds., New York: McGraw-Hill Book Company, 1986. |
| [15] |
A. E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39–56. https://doi.org/10.1007/BF01404755 doi: 10.1007/BF01404755
|
| [16] |
R. Yang, Y.-H. Lee, I. Sim, Bifurcation of nodal radial solutions for a prescribed mean curvature problem on an exterior domain, J. Differ. Equations, 268 (2020), 4464–4490. https://doi.org/10.1016/j.jde.2019.10.035 doi: 10.1016/j.jde.2019.10.035
|
Zhongzi Zhao, Meng Yan. Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain[J]. AIMS Mathematics, 2023, 8(9): 20654-20664. doi: 10.3934/math.20231052
DownLoad: