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Topological Methods in Nonlinear Analysis

Singular reaction diffusion equations where a parameter influences the reaction term and the boundary conditions
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Singular reaction diffusion equations where a parameter influences the reaction term and the boundary conditions

Authors

  • Nalin Fonseka
  • Amila Muthunayake
  • Ratnasingham Shivaji
  • Byungjae Son

Keywords

Boundary value problems, singular problems, positive solutions

Abstract

We analyse positive solutions to the steady state reaction diffusion equation: \begin{equation*} \label{1.11} \begin{cases} -u''=\lambda h(t) f(u) \quad \text{in } (0,1), \\ -du'(0)+\mu(\lambda) u(0)=0,\\ u'(1)+\mu(\lambda) u(1)=0, \end{cases} \end{equation*} where $\lambda> 0$ is a parameter, $d\geq 0$ is a constant, $f \in C^2([0,\infty),\mathbb{R}) $ is an increasing function which is sublinear at infinity $\Big (\lim\limits_{s \rightarrow \infty}{f(s)}/{s}=0\Big)$, $h \in C^1((0,1],(0,\infty))$ is a nonincreasing function with $h_1:=h(1)> 0$ and there exist constants $d_0> 0$, $\alpha \in [0,1)$ such that $h(t)\leq {d_0}/{t^\alpha}$ for all $t \in (0,1]$, and $\mu \in C([0,\infty),[0,\infty))$ is an increasing function such that $\mu(0)\geq 0$. We consider three cases of $f$, namely, $f(0)=0$, $f(0)> 0$ and $f(0)< 0$. We will discuss existence and multiplicity results via the method of sub-supersolutions. Further, we will establish uniqueness results for $\lambda\approx 0$ and $\lambda\gg 1$.

References

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

D. Butler, E. Ko and R. Shivaji, Alternate steady states for classes of reaction diffusion models on exterior domains, Discrete Contin. Dyn. Syst. Ser. S 7 (2014), No. 6, 1181–1191.

A. Castro, J.B. Garner and R. Shivaji, Existence results for classes of sublinear semipositone problems, Results Math. 23 (1993), 214–220.

A. Castro, M. Hassanpour and R. Shivaji, Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity, Comm. Partial Differential Equations 20 (1995), No. 11–12, 1927–1936.

A. Castro, L. Sankar and R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl. 394 (2012), No. 1, 432–437.

K.D. Chu, D.D. Hai and R. Shivaji, Uniqueness of positive radial solutions for infinite semipositone p-Laplacian problems in exterior domains, J. Math. Anal. Appl. 472 (2019), No. 1, 510–525.

J. Cronin, J. Goddard and R. Shivaji, Effects of patch-matrix composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol. 81 (2019), No. 10, 3933–3975.

R. Dhanya, E. Ko and R. Shivaji, A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 424 (2015), No. 1, 598–612.

R. Dhanya, R. Shivaji and B. Son, A three solution theorem for a singular differential equation with nonlinear boundary conditions, Topol. Methods Nonlinear Anal. 54 (2019), No. 2A, 445–457.

N. Fonseka, R. Shivaji, B. Son and K. Spetzer, Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition, J. Math. Anal. Appl. 476 (2019), No. 2, 480–494.

J. Goddard II, Q. Morris, S. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl. 1 (2018), Art. No. 170.

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

E.K. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains, Differential Integral Equations 24 (2011), 861–875.

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

M.A. Rivas and S. Robinson, Eigencurves for linear elliptic equations, ESAIM Control Optim. Calc. Var. 25 (2019), Art. 45, 25 pp.

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.), vol. 109, 1987, pp. 561–566.

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Published

2021-02-08

How to Cite

1.
FONSEKA, Nalin, MUTHUNAYAKE, Amila, SHIVAJI, Ratnasingham and SON, Byungjae. Singular reaction diffusion equations where a parameter influences the reaction term and the boundary conditions. Topological Methods in Nonlinear Analysis. Online. 8 February 2021. Vol. 57, no. 1, pp. 221 - 242. [Accessed 3 July 2025].
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