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


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.


Principal eigenvalues Nonvariational degenerate quasilinear problems Comparison principles Fredholm alternative 

Mathematics Subject Classification

35J92 35B51 



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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