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Revista Matemática Complutense

, Volume 30, Issue 2, pp 313–334 | Cite as

Dirichlet problems for the p-Laplacian with a convection term

  • Jorge García-Melián
  • José C. Sabina de LisEmail author
  • Peter Takáč
Article

Abstract

We consider the nonlinear Dirichlet boundary value problem
in a bounded domain \(\Omega \subset \mathbb {R}^N\) with smooth boundary \(\partial \Omega \), where \(\Delta _p u\mathop {=}\limits ^{\mathrm{{def}}}\mathrm {div} (|\nabla u|^{p-2} \nabla u)\) with \(1< p < \infty \), \(\lambda \in \mathbb {R}\), and \(h\in L^\infty (\Omega )\). The term \(B(x,\nabla u)\) is a continuous function assumed to be also homogeneous of degree \((p-1)\) and odd with respect to the second variable; \(B(x, \varvec{\eta }) = (\mathbf {a}(x)\cdot \varvec{\eta }) |\varvec{\eta }|^{p-2}\) being a canonical example with a given vector field \(\mathbf {a}\in [ C(\overline{\Omega }) ]^N\), for \((x, \varvec{\eta })\in \Omega \times \mathbb {R}^N\). For the corresponding eigenvalue problem obtained by setting \(h\equiv 0\), we show existence, simplicity, and isolation of the principal eigenvalue \(\lambda _1\) (\(\lambda _1 > 0\)). When \(h\not \equiv 0\) and \(-\infty< \lambda < \lambda _1\), we prove that there exists a weak solution \(u\in W_0^{1,p}(\Omega )\) to problem (P); this solution is unique provided \(\lambda < 0\) (without any further assumptions). When \(h\ge 0\), \(h\not \equiv 0\), and \(0\le \lambda < \lambda _1\), we show that the solution is positive and also unique.

Keywords

Principal eigenvalues Nonvariational degenerate quasilinear problems Comparison principles Fredholm alternative 

Mathematics Subject Classification

35J92 35B51 

Notes

Acknowledgements

Supported by Spanish Ministerio de Ciencia e Innovación and Ministerio de Economía y Competitividad under grant reference MTM2011-27998.

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Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  • Jorge García-Melián
    • 1
  • José C. Sabina de Lis
    • 1
    Email author
  • Peter Takáč
    • 2
  1. 1.Departamento de Análisis Matemático and IUdEAUniversidad de La LagunaLa LagunaSpain
  2. 2.Institut für MathematikUniversität RostockRostockGermany

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