Differential and Integral Equations

On the existence and multiplicity of positive solutions of the $p$-Laplacian separated boundary value problem

A. K. Ben-Naoum and C. De Coster

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Using the lower and upper solutions method together with degree theory, we study the existence and multiplicity of positive solutions for the problem $$ (\varphi_{p}(u'))'+f(t,u)=0,\ \ a_{1}\varphi_{p}(u(a))-a_{2} \varphi_{p}(u'(a))=0,\ b_{1}\varphi_{p}(u(b))+b_{2} \varphi_{p}(u'(b))=0, $$ where $\varphi_{p} (s):=|s|^{p-2}s, \,p>1$, $a_1,b_1\in\Bbb R$, $a_2,b_2\in\Bbb R^+$, $a_1^2+a_2^2>0$, $b_1^2+b_2^2>0.$ The function $f$ satisfies assumptions related to the classically called sublinear, superlinear, subsuperlinear, or supersublinear cases. Our results improve the recent ones of L.H. Erbe-H. Wang ([21]) and L.H. Erbe-S. Hu-H. Wang ([20]).

Article information

Differential Integral Equations, Volume 10, Number 6 (1997), 1093-1112.

First available in Project Euclid: 1 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 34B15: Nonlinear boundary value problems 35B32: Bifurcation [See also 37Gxx, 37K50]


Ben-Naoum, A. K.; De Coster, C. On the existence and multiplicity of positive solutions of the $p$-Laplacian separated boundary value problem. Differential Integral Equations 10 (1997), no. 6, 1093--1112. https://projecteuclid.org/euclid.die/1367438221

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