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

Article

Abstract

In this work we study the existence of critical points of an abstract \(C^1\) functional J defined in a reflexive Banach space X. This functional is of the form
$$\begin{aligned} J(u)=\dfrac{1}{p}E(u)-\dfrac{1}{r}A(u)-\dfrac{1}{q}B(u), \end{aligned}$$
with EAB positive-homogeneous indefinite functional of degree pqr respectively and \(1<p<q<r\). 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
$$\begin{aligned} \left\{ \begin{array}{lllll} -\Delta _p u+V(x)u^{p-1} =0&{} \text { in } \Omega ;\\ |\nabla u|^{p-2}\dfrac{\partial u}{\partial \nu }=\lambda a(x)u^{r-1} +b(x)u^{q-1}&{}\text { on }\partial \Omega ; \\ u>0 \text { in }\Omega , \end{array} \right. \end{aligned}$$
with given functions abV possibly indefinite and \(1<r<p<q\). We also apply our abstract result for a concave–convex quasilinear problem associated to the p-bilaplacian.

Keywords

Concave–convex problem Nehari manifold Dirichlet–Steklov boundary condition Elliptic problem Non-coerciveness p-Laplacian p-bilaplacian Indefinite weights 

Mathematics Subject Classification

35J20 35J70 35P05 35P30 

Notes

Acknowledgements

This work was partilly carried out while the first author was visiting the IMSP of the Université d’Abomey Calavi (Porto-Novo) and also while the second author was visiting the Université du Littoral Côte d’Opale (ULCO) and the Université Libre de Bruxelles (ULB). We would like to express our gratitude to those institutions.

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.LMPA Joseph Liouville, FR CNRS Math. 2956Université Lille Nord de FranceCalaisFrance
  2. 2.Département de Mathématiques, Faculté des Sciences et Techniques Institut de Mathématiques et de Sciences PhysiquesUniversité d’Abomey-CalaviPorto-NovoBenin

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