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Positive solutions to classes of infinite semipositone \((p,q)\)-Laplace problems with nonlinear boundary conditions. (English) Zbl 1483.34043

In this interesting paper the authors study the existence, multiplicity and nonexistence of positive solutions for the one-dimensional \((p,q)\)-Laplacian problems: \begin{gather*} -(\varphi(u'))'=\lambda h(t)f(u),\qquad t\in (0,1),\\ u(0)=0=au'(1)+g(\lambda,u(1))u(1), \end{gather*} where \(\lambda>0\), \(a\geq 0\), \(\varphi(s)=|s|^{p-2}s+|s|^{q-2}s\) with \(1<p<q<+\infty\), \(h\in C((0,1),(0,+\infty))\), and \(f\in C((0,+\infty),\mathbb{R})\) may have a singularity at 0 of repulsive type. The proofs are based on a classical Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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