Singular equations with variable exponents and concave-convex nonlinearities

Abstract

<p style='text-indent:20px;'>We consider a nonlinear anisotropic Dirichlet problem with a reaction that has the combined effects of three distinct nonlinearities: a parametric singular term, a parametric "concave" term (the parameter <inline-formula><tex-math id="M1">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is the same in both) and a nonparametric "convex" perturbation. So, the problem is a singular version of the well known "concave-convex" problem. We prove an existence and multiplicity result which is global in the parameter <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> (a bifurcation-type theorem). We also indicate some small improvements in the case of <inline-formula><tex-math id="M3">\begin{document}$ (p(z),q(z)) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p(z) $\end{document}</tex-math></inline-formula>-equations.</p>