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2 results

POSITIVE SOLUTIONS FOR NON-RESONANT SINGULAR BOUNDARY-VALUE PROBLEMS WITH A LINEAR TERM
Haishen Lü, Donal O’Regan, Ravi P. Agarwal
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 50 / Issue 1 / February 2007

Published online by Cambridge University Press:

09 February 2007, pp. 217-228
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This paper presents new existence results for the singular boundary-value problem
\begin{gather*} -u''+p(t)u=f(t,u),\quad t\in(0,1),\\ u(0)=0=u(1). \end{gather*}
In particular, our nonlinearity $f$ may be singular at $t=0,1$ and $u=0$.

Existence and Multiplicity of Positive Solutions for Singular Semipositone p-Laplacian Equations
Ravi P. Agarwal, Daomin Cao, Haishen Lü, Donal O'Regan
Journal:

Canadian Journal of Mathematics / Volume 58 / Issue 3 / 01 June 2006

Published online by Cambridge University Press:

20 November 2018, pp. 449-475

Print publication:

01 June 2006
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Positive solutions are obtained for the boundary value problem
$$\left\{ _{u\left( 0 \right)\,=\,u\left( 1 \right)\,=\,0}^{-{{\left( {{\left| {{u}'} \right|}^{p-2}}{u}' \right)}^{\prime }}\,=\,\lambda f\left( t,\,u \right),\,t\,\in \,\left( 0,\,1 \right),\,p\,>\,1} \right.$$
Here $f(t,u)\ge -M$, ($M$ is a positive constant) for $(t,u)\in [0,1]\times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.