Positive solutions for nonlinear problems involving the one-dimensional ϕ -Laplacian

MILNE, LEANDRO; KAUFMANN, URIEL; KAUFMANN, URIEL; MILNE, LEANDRO

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2018 vol. 461 p. 24 - 37

0022-247X

Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left(\Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be thedifferential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right)^{\prime}+r\left( x\right) \phi\left( u\right) $, where $\phi:\mathbb{R\rightarrow R}$ is an odd increasing homeomorphism and $0\leq r\inL^{1}\left( \Omega\right) $. We study the existence of \textit{positive}solutions for problems of the form%\[\left\{\begin{array}[c]{ll}%\mathcal{L}u=\lambda m\left( x\right) f\left( u\right) & \text{in }%\Omega,\\u=0 & \text{on }\partial\Omega,\end{array}\right.\]where $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ is acontinuous function which is, roughly speaking, sublinear with respect to$\phi$. Our approach combines the sub and supersolution method with someestimates on related nonlinear problems. We point out that our results are neweven in the cases $r\equiv0$ and/or $m\geq0$.