### AIMS Mathematics

2021, Issue 8: 8949-8958. doi: 10.3934/math.2021519
Research article

# Positive radial solutions of p-Laplace equations on exterior domains

• Received: 23 March 2021 Accepted: 07 June 2021 Published: 15 June 2021
• MSC : 35J25, 35J60, 47H11, 47N20

• This paper deals with the existence of positive radial solutions of the $p$-Laplace equation

$\left\{\begin{array}{ll} -\Delta_p\,u= K(|x|)\,f(u)\,,\quad x\in\Omega\,,\qquad\qquad\\[6pt] \frac{\partial u}{\partial n}=0\,,\qquad x\in\partial\Omega,\\[6pt] \lim_{|x|\to\infty}u(x)=0\,, \end{array}\right.$

where $\Omega = \{x\in {\mathbb{R}}^N:\, |x| > r_0\}$, $N\ge 2$, $1 < p < N$, $K: [r_0, \, \infty)\to {\mathbb{R}}^+$ is continuous and $0 < \int_{r_0}^{\infty}r^{N-1}K(r)\, dr < \infty$, $f\in C({\mathbb{R}}^+, \, {\mathbb{R}}^+)$. Under the inequality conditions related to the asymptotic behaviour of $f(u)/u^{p-1}$ at $0$ and infinity, the existence results of positive radial solutions are obtained. The discussion is based on the fixed point index theory in cones.

Citation: Yongxiang Li, Mei Wei. Positive radial solutions of p-Laplace equations on exterior domains[J]. AIMS Mathematics, 2021, 6(8): 8949-8958. doi: 10.3934/math.2021519

### Related Papers:

• This paper deals with the existence of positive radial solutions of the $p$-Laplace equation

$\left\{\begin{array}{ll} -\Delta_p\,u= K(|x|)\,f(u)\,,\quad x\in\Omega\,,\qquad\qquad\\[6pt] \frac{\partial u}{\partial n}=0\,,\qquad x\in\partial\Omega,\\[6pt] \lim_{|x|\to\infty}u(x)=0\,, \end{array}\right.$

where $\Omega = \{x\in {\mathbb{R}}^N:\, |x| > r_0\}$, $N\ge 2$, $1 < p < N$, $K: [r_0, \, \infty)\to {\mathbb{R}}^+$ is continuous and $0 < \int_{r_0}^{\infty}r^{N-1}K(r)\, dr < \infty$, $f\in C({\mathbb{R}}^+, \, {\mathbb{R}}^+)$. Under the inequality conditions related to the asymptotic behaviour of $f(u)/u^{p-1}$ at $0$ and infinity, the existence results of positive radial solutions are obtained. The discussion is based on the fixed point index theory in cones.

 [1] J. Santanilla, Existence and nonexistence of positive radial solutions of an elliptic Dirichlet problem in an exterior domain, Nonlinear Anal., 25 (1995), 1391–1399. doi: 10.1016/0362-546X(94)00255-G [2] Y. H. Lee, Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains, Differ. Integral Equ, 13 (2000), 631–648. [3] Y. H. Lee, A multiplicity result ofpositive radial solutions for a multiparameter elliptic system on an exterior domain, Nonlinear Anal., 45 (2000), 597–611. [4] R. Stanczy, Decaying solutions for sublinear elliptic equations in exterior domains, Topol. Method. Nonlinear Anal., 14 (1999), 363–370. doi: 10.12775/TMNA.1999.039 [5] R. Stanczy, Positive solutions for superlinear elliptic equations, J. Math. Anal. Appl., 283 (2003), 159–166. doi: 10.1016/S0022-247X(03)00265-8 [6] R. Precup, Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl., 352 (2009), 48–56. doi: 10.1016/j.jmaa.2008.01.097 [7] Y. Li, H. Zhang, Existence of positive radial solutions for the elliptic equations on an exterior domain, Ann. Polon. Math., 116 (2016), 67–78. [8] Z. Liu, F. Li, Multiple positive solutions of nonlinear boundary problems, J. Math. Anal. Appl., 203 (1996), 610–625. doi: 10.1006/jmaa.1996.0400 [9] L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Model., 32 (2000), 529–539. doi: 10.1016/S0895-7177(00)00150-3 [10] Y. Li, Abstract existence theorems of positive solutions for nonlinear boundary value problems, Nonlinear Anal., 57 (2004), 211–227. doi: 10.1016/j.na.2004.02.010 [11] L. M. Dai, Existence and nonexistence of subsolutions for augmented Hessian equations, DCDS, 40 (2020), 579–596. doi: 10.3934/dcds.2020023 [12] L. M. Dai, J. G. Bao, On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains, Front. Math. China, 6 (2011), 221–230. doi: 10.1007/s11464-011-0109-x [13] W. Walter, Sturm-Liouville theory for the radial $\Delta_p$-operator, Math. Z., 227 (1998), 175–185. doi: 10.1007/PL00004362 [14] J. F. Bonder, J. P. Pinasco, Asymptotic behavior of the eigenvalues of the one-dimensional weighted p-Laplace operator, Ark. Mat., 41 (2003), 267–280. doi: 10.1007/BF02390815 [15] M. Lucia, S. Prashanth, Simplicity of principal eigenvalue for p-Laplace operator with singular indefinite weight, Arch. Math., 86 (2006), 79–89. doi: 10.1007/s00013-005-1512-x [16] K. Deimling, Nonlinear functional analysis, New York: Springer-Verlag, 1985. [17] D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, New York: Academic Press, 1988.
• ##### Reader Comments
• © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.427 1.6

## Metrics

Article views(339) PDF downloads(48) Cited by(0)

Article outline

## Other Articles By Authors

• On This Site
• On Google Scholar

/

DownLoad:  Full-Size Img  PowerPoint