Electron. J. Differential Equations, Vol. 2018 (2018), No. 189, pp. 1-12.

Exactness of the number of positive solutions to a singular quasilinear problem

Giovanni Anello, Luca Vilasi

Abstract:
We study the exact multiplicity of positive solutions to the one-dimensional Dirichlet problem
$$\displaylines{ -(|u'|^{p-2}u')' = \lambda u^{s-1} - \mu u^{r-1} \quad \text{in } ]0,1[\cr u(0) = u(1) = 0, }$$
where $r\in]0,1[$, $p\in]1,+\infty[$, $r<s<p$ and $\lambda,\mu\in]0,+\infty[$. We shed light, in particular, on the case $r\in]0,\min\{s, p/(p+1)\}[$, completely determining the bifurcation diagram and solving some related open problems. Our approach relies upon quadrature methods.

Submitted April 19, 2018. Published November 20, 2018.
Math Subject Classifications: 34B08, 34B16, 34B18.
Key Words: Exactness; singular problem; positive solution; quadrature method.

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Giovanni Anello
Department of Mathematics and Computer Sciences
Physical Sciences and Earth Sciences
University of Messina
Viale F. Stagno d'Alcontres 31
98166 Messina, Italy
email: ganello@unime.it
Luca Vilasi
Department of Mathematics and Computer Sciences
Physical Sciences and Earth Sciences
University of Messina
Viale F. Stagno d'Alcontres 31
98166 Messina, Italy
email: lvilasi@unime.it

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