# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5153-5166. doi: 10.3934/dcds.2013.33.5153

## Multiplicity results for classes of singular problems on an exterior domain

 1 Department of Mathematics and computer science, McDaniel College, Westminster, MD 21157, United States 2 Department of Mathematics Education, Pusan National University, Busan, 609-735, South Korea 3 Department of Mathematics & Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, United States

Received  August 2011 Revised  March 2012 Published  May 2013

We study radial positive solutions to the singular boundary value problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda K(|x|)\frac{f(u)}{u^\beta} \quad \mbox{in}~ \Omega,\\ ~~~u(x) = 0 \qquad \qquad \qquad \qquad~~\mbox{if}~|x|=r_0,\\ ~~~u(x) \rightarrow 0 \qquad\qquad \qquad \mbox{if}~|x|\rightarrow \infty, \end{cases} \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $1 < p < N, N >2, \lambda > 0, 0 \leq \beta <1 ,\Omega= \{ x \in \mathbb{R}^{N} : |x| > r_0 \}$ and $r_0 >0$. Here $f:[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0$ and $K \in C( (r_0, \infty),(0, \infty) )$ is such that $\int_{r_0}^{\infty} r^\mu K(r) dr < \infty,$ for some $\mu > p-1$. We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $\alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-super solutions.
Citation: Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5153-5166. doi: 10.3934/dcds.2013.33.5153
##### References:
 [1] S. Cui, Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems,, Nonlinear Analysis, 41 (2000), 149.  doi: 10.1016/S0362-546X(98)00271-5.  Google Scholar [2] D. Jiang and W. Gao, Upper and lower solution method and a singular boundary value problem for one-dimensional p-Laplacian,, J. Math. Anal. Appl., 252 (2000), 631.  doi: 10.1006/jmaa.2000.7012.  Google Scholar [3] L. Haishen and D. O'Regan, A general existence theorem for singular equation $(\varphi_p(y'))' + f(t,y)=0$,, Math. Inequal. Appl., 5 (2002), 69.  doi: 10.7153/mia-05-09.  Google Scholar [4] R. Kajikiya, Y.-H. Lee and I. Sim, One-dimensional p-Laplacian with a strong singular indefinite weight, I. Eigenvalue,, J. Differential Equations, 244 (2008), 1985.  doi: 10.1016/j.jde.2007.10.030.  Google Scholar [5] C. Kim, E. K. Lee and Yong-Hoon Lee, Existence of the second positive radial solution for a p-Laplacian problem,, J. Comput. Appl. Math., 235 (2011), 3743.  doi: 10.1016/j.cam.2011.01.020.  Google Scholar [6] E. Ko, E. K. Lee and R. Shivaji, Multiplicity results for classes of infinite positone problems,, Z. Anal. Anwend., 30 (2011), 305.  doi: 10.4171/ZAA/1436.  Google Scholar [7] E. K. Lee and Y.-H. Lee, A global multiplicity result for two-point boundary value problems of p-Laplacian systems,, Sci. China Math., 53 (2010), 967.  doi: 10.1007/s11425-010-0088-5.  Google Scholar [8] E. K. Lee, R. Shivaji and J. Ye, Classes of infinite semipositone systems,, Proc. Roy. Soc. Edinburgh. Sect. A, 139 (2009), 853.  doi: 10.1017/S0308210508000255.  Google Scholar [9] Do O J. Marcos, S. Lorca S and J. Sanchez et al., Positive radial solutions for some quasilinear elliptic systems in exterior domains,, Comm. Pure Appl. Anal., 5 (2006), 571.  doi: 10.3934/cpaa.2006.5.571.  Google Scholar

show all references

##### References:
 [1] S. Cui, Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems,, Nonlinear Analysis, 41 (2000), 149.  doi: 10.1016/S0362-546X(98)00271-5.  Google Scholar [2] D. Jiang and W. Gao, Upper and lower solution method and a singular boundary value problem for one-dimensional p-Laplacian,, J. Math. Anal. Appl., 252 (2000), 631.  doi: 10.1006/jmaa.2000.7012.  Google Scholar [3] L. Haishen and D. O'Regan, A general existence theorem for singular equation $(\varphi_p(y'))' + f(t,y)=0$,, Math. Inequal. Appl., 5 (2002), 69.  doi: 10.7153/mia-05-09.  Google Scholar [4] R. Kajikiya, Y.-H. Lee and I. Sim, One-dimensional p-Laplacian with a strong singular indefinite weight, I. Eigenvalue,, J. Differential Equations, 244 (2008), 1985.  doi: 10.1016/j.jde.2007.10.030.  Google Scholar [5] C. Kim, E. K. Lee and Yong-Hoon Lee, Existence of the second positive radial solution for a p-Laplacian problem,, J. Comput. Appl. Math., 235 (2011), 3743.  doi: 10.1016/j.cam.2011.01.020.  Google Scholar [6] E. Ko, E. K. Lee and R. Shivaji, Multiplicity results for classes of infinite positone problems,, Z. Anal. Anwend., 30 (2011), 305.  doi: 10.4171/ZAA/1436.  Google Scholar [7] E. K. Lee and Y.-H. Lee, A global multiplicity result for two-point boundary value problems of p-Laplacian systems,, Sci. China Math., 53 (2010), 967.  doi: 10.1007/s11425-010-0088-5.  Google Scholar [8] E. K. Lee, R. Shivaji and J. Ye, Classes of infinite semipositone systems,, Proc. Roy. Soc. Edinburgh. Sect. A, 139 (2009), 853.  doi: 10.1017/S0308210508000255.  Google Scholar [9] Do O J. Marcos, S. Lorca S and J. Sanchez et al., Positive radial solutions for some quasilinear elliptic systems in exterior domains,, Comm. Pure Appl. Anal., 5 (2006), 571.  doi: 10.3934/cpaa.2006.5.571.  Google Scholar
 [1] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [2] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [3] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [4] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442 [5] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [6] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [7] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [8] Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117 [9] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 [10] Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452 [11] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [12] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [13] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [14] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [15] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [16] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [17] Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274 [18] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [19] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [20] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

2019 Impact Factor: 1.338