A three solution theorem for a singular differential equation with nonlinear boundary conditions

Rajendran Dhanya, Ratnasingham Shivaji, Byungjae Son

DOI: http://dx.doi.org/10.12775/TMNA.2019.044


We study positive solutions to singular boundary value problems of the form: \begin{equation*} \begin{cases} -u'' = h(t) \dfrac{f(u)}{u^\alpha} &\text{for } t \in (0,1), \\ u(0) = 0, \\ u'(1) + c(u(1)) u(1) = 0,\hidewidth \end{cases} \end{equation*} where $0< \alpha< 1$, $h\colon(0,1]\rightarrow(0,\infty)$ is continuous such that $h(t)\leq {d}/{t^\beta}$ for some $d> 0$ and $\beta\in[0,1-\alpha)$ and $c\colon [0,\infty)\rightarrow [0,\infty)$ is continuous such that $c(s)s$ is nondecreasing. We assume that $f\colon[0,\infty)\rightarrow(0,\infty)$ is continuously differentiable such that $[(f(s)-f(0))/s^\alpha ]+\tau s$ is strictly increasing for some $\tau\geq 0$ for $s\in(0,\infty)$. When there exists a pair of sub-supersolutions $(\psi,\phi)$ such that $0\leq \psi\leq\phi$, we first establish a minimal solution $\underline u$ and a maximal solution $\overline u$ in $[\psi,\phi]$. When there exist two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where $0\leq \psi_1 \leq \psi_2 \leq \phi_1$, $\psi_1 \leq \phi_2 \leq \phi_1$ with $\psi_2\not \leq \phi_2$, and $\psi_2$, $\phi_2$ are not solutions, we next establish the existence of at least three solutions $u_1$, $u_2$ and $u_3$ satisfying $u_1\in [\psi_1,\phi_2], u_2\in [\psi_2,\phi_1]$ and $u_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.


Singular boundary value problem; nonlinear boundary conditions; three solution theorem

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H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709.

K. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, J. Nonlinear Anal. 5 (1981), 475–486.

D. Butler, E. Ko, E.K. Lee and R. Shivaji, Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions, Commun. Pure Appl. Anal. 13 (2014), 2713–2731.

R.S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2004.

R.S. Cantrell and C. Cosner, Density dependent behavior at habitat boundaries and the allee effect, Bull. Math. Biol. 69 (2007), 2339–2360.

D. A. Frank-Kamenetskiı̆, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, 1969.

J. Goddard II, E.K. Lee and R. Shivaji, Population models with diffusion, strong allee effect, and nonlinear boundary conditions, Nonlinear Anal. 74 (2011), 6202–6208.

E.K. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl. 381 (2011), 732–741.

E.K. Lee, R. Shivaji and B. Son, Positive radial solutions to classes of singular problems on the exterior domain of a ball, J. Math. Anal. Appl. 434 (2016), 1597–1611.

R. Dhanya, E. Ko and R. Shivaji, A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 424 (2015), 598–612.

R. Dhanya, E. Ko and R. Shivaji, A three solution theorem for a two-point singular boundary value problem with an unbounded weight, Electron. J. Differ. Equ. Conf. 23 (2016), 131–138.

R. Dhanya, Q. Morris and R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl. 434 (2016), 1533–1548.

J. Giacomoni, I. Schindler and P. Takác̆, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007), 117–158.

F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J. 31 (1982), 213–221.

M.H. Protter and H.F. Weinberger, Maximum principles in differential equations, Springer–Verlag, 1984

N.N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, 1935.

R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications, Lecture Notes in Pure and Applied Mathematics (V. Lakshmikantham, ed.) 109 (1987), 561–566.

Y.B. Zeldovich, G.I. Barenblatt, V.B. Librovich and G.M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, 1985.


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