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Elliptic Problems with Nonhomogeneous Differential Operators and Multiple Solutions

  • Dumitru Motreanu
  • Patrick Winkert
Chapter

Abstract

The present survey aims to report on recent advances in the study of non-linear elliptic problems whose differential part is expressed by a general operator in divergence form. The pattern of such differential operator is the p-Laplacian Δ p with \(1<p<+\infty\). More general operators can be considered, possibly having completely different properties, for instance not satisfying any homogeneity requirement. A major objective of our work is to provide existence theorems of multiple solutions for boundary value problems governed by such general operators. In this direction, a three nontrivial solutions theorem is presented. In the case of problems determined by the p-Laplacian, we give a theorem ensuring the existence of at least four nontrivial solutions. Moreover, a complete sign information is available: two positive solutions, a negative solution and a nodal (sign-changing) solution. Finally, we provide a theorem guaranteeing the existence of a positive solution for a problem involving the \((p,q)\)-Laplacian operator \(\Delta_p+\Delta_q\), with \(1<q<p\), and a nonlinearity depending on the solution and its gradient.

Keywords

Nonlinear elliptic boundary value problem Nonhomogeneous differential operator p-Laplacian Eigenvalue problem Multiple solutions Critical point Minimizer Regularity Maximum principle 

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Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de PerpignanPerpignanFrance
  2. 2.Technische Universität BerlinInstitut für MathematikBerlinGermany

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