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 and Continuous Dynamical Systems, 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), 149-176. doi: 10.1016/S0362-546X(98)00271-5.

[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), 631-648. doi: 10.1006/jmaa.2000.7012.

[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), 69-78. doi: 10.7153/mia-05-09.

[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), 1985-2019. doi: 10.1016/j.jde.2007.10.030.

[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), 3743-3750. doi: 10.1016/j.cam.2011.01.020.

[6]

E. Ko, E. K. Lee and R. Shivaji, Multiplicity results for classes of infinite positone problems, Z. Anal. Anwend., 30 (2011), 305-318. doi: 10.4171/ZAA/1436.

[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), 967-984. doi: 10.1007/s11425-010-0088-5.

[8]

E. K. Lee, R. Shivaji and J. Ye, Classes of infinite semipositone systems, Proc. Roy. Soc. Edinburgh. Sect. A, 139 (2009), 853-865. doi: 10.1017/S0308210508000255.

[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), 571-581. doi: 10.3934/cpaa.2006.5.571.

show all references

References:
[1]

S. Cui, Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems, Nonlinear Analysis, 41 (2000), 149-176. doi: 10.1016/S0362-546X(98)00271-5.

[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), 631-648. doi: 10.1006/jmaa.2000.7012.

[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), 69-78. doi: 10.7153/mia-05-09.

[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), 1985-2019. doi: 10.1016/j.jde.2007.10.030.

[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), 3743-3750. doi: 10.1016/j.cam.2011.01.020.

[6]

E. Ko, E. K. Lee and R. Shivaji, Multiplicity results for classes of infinite positone problems, Z. Anal. Anwend., 30 (2011), 305-318. doi: 10.4171/ZAA/1436.

[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), 967-984. doi: 10.1007/s11425-010-0088-5.

[8]

E. K. Lee, R. Shivaji and J. Ye, Classes of infinite semipositone systems, Proc. Roy. Soc. Edinburgh. Sect. A, 139 (2009), 853-865. doi: 10.1017/S0308210508000255.

[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), 571-581. doi: 10.3934/cpaa.2006.5.571.

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