On abstract indefinite concave–convex problems and applications to quasilinear elliptic equations

In this work we study the existence of critical points of an abstractC1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} functional J defined in a reflexive Banach space X. This functional is of the form J(u)=1pE(u)-1rA(u)-1qB(u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(u)=\dfrac{1}{p}E(u)-\dfrac{1}{r}A(u)-\dfrac{1}{q}B(u), \end{aligned}$$\end{document}with E, A, B positive-homogeneous indefinite functional of degree p, q, r respectively and 1<p<q<r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q<r$$\end{document}. The critical points are found by minimization along several subsets of the Nehari manifold associated to J. We apply these results to various quasilinear elliptic problems, as for instance, the following p-laplacian concave–convex problem with Steklov boundary conditions on a bounded regular domain -Δpu+V(x)up-1=0inΩ;|∇u|p-2∂u∂ν=λa(x)ur-1+b(x)uq-1on∂Ω;u>0inΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{lllll} -\Delta _p u+V(x)u^{p-1} =0&{} \text { in } \Omega ;\\ |\nabla u|^{p-2}\dfrac{\partial u}{\partial \nu }=\lambda a(x)u^{r-1} +b(x)u^{q-1}&{}\text { on }\partial \Omega ; \\ u>0 \text { in }\Omega , \end{array} \right. \end{aligned}$$\end{document}with given functions a, b, V possibly indefinite and 1<r<p<q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<r<p<q$$\end{document}. We also apply our abstract result for a concave–convex quasilinear problem associated to the p-bilaplacian.