The existence of positive solutions for the singular two-point boundary value problem

Yanmin Niu, Baoqiang Yan


In this paper, we consider the following boundary value problem: $$ \begin{cases} ((-u'(t))^n)'=nt^{n-1}f(u(t)) &\text{for }0< t< 1,\\ u'(0)=0,\quad u(1)=0, \end{cases} $$% where $n> 1$. Using the fixed point theory on a cone and approximation technique, we obtain the existence of positive solutions in which $f$ may be singular at $u=0$ or $f$ may be sign-changing.


Singular differential equation; two-point boundary value problem; fixed point theorem; positive solution

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R.P. Agarwal and D. O’Regan, Singular boundary value problems for superlinear second ordinary and delay differential equations, J. Differential Equations 130 (1996), 335–355.

R.P. Agarwal and D. O’Regan, Nonlinear superlinear singular and nonsingular second order boundary value problems, J. Differential Equations 143 (1998), 60–95.

G. Dai, Eigenvalue, bifurcation, existence and nonexistence of solutions for Monge–Ampére equations, Analysis of PDEs, arXiv:1007.3013 (2010).

P. Delanoe, Radially symmetric boundary value problems for real and complex elliptic Monge–Ampère equations, J. Differential Equations 58 (1985), 318–344.

J.V.A. Goncalves and C.A.P. Santos, Classical solutions of singular Monge–Ampère equation a ball, J. Math. Anal. Appl. 305 (2005), 240–252.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.

C. Gutierrez, The Monge–Ampère Equation, Birkhäuser, Basel, 2000.

S. Hu and H. Wang, Convex Solutions of Boundary Value Problems Arising from Monge–Ampère Equations, Discrete Contin. Dyn. Syst. 16 (2006), 705–720.

N.D. Kutev, Nontrivial solutions for the equations of Monge–Ampère type, J. Math. Anal. Appl. 132 (1988), 424–433.

A.C. Lazer and P.J. McKenna, On singular boundary value problems for the Monge–Ampère operator, J. Math. Anal. Appl. 197 (1996), 341–362.

P.L. Lions, Two remarks on Monge–Ampère equations, Ann. Mat. Pura Appl. 142 (1985), 263–275.

L. Ma and B. Liu, Symmetry results for classical solutions of Monge–Ampère systems on bounded planar domains, J. Math. Anal. Appl. 369 (2010), 678–685.

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation, J. Math. Anal. Appl. 340 (2008), 1226–1234.

A. Mohammed, Singular boundary value problems for the Monge–Ampère equation, Nonlinear Anal. 70 (2009), 457–464.

J. Wang, W. Gao and Z. Lin, Boundary value problems for general second order equations and similarity solutions to the rayleing problem, Tohoku Math. J. 47 (1995), 327–344.

F. Wang and Y. An, Triple nontrivial radial convex solutions of systems of Monge–Ampère equations, Appl. Math. Lett. 25 (2012), 88–92.

H. Wang, Radial convex solutions of boundary value problems for systems of Monge–Ampère equations, Anal. Partial Differential Equations, arXiv:1008.4614 (2010).

H. Wang, Convex solutions of systems arising from Monge–Ampère equations, Electron. J. Qual. Theory Differ. Equ. 26 (2009), 1–8.

G. Yang, Positive solutions of singular Dirichlet boundary value problems with signchanging nonlinearities, Comput. Math. Appl. 51 (2006), 1463–1470.

Z. Zhang and K. Wang, Existence and non-existence of solutions for a class of Monge–Ampère equations, J. Differential Equations 246 (2009), 2849–2875.


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