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Mountain pass type solutions for quasilinear elliptic equations

  • Ph. Clément
  • M. García-Huidobro
  • R. Manásevich
  • K. Schmitt
Original article

Abstract.

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem

\((D)\quad \left \{\begin{array}{rcll} -{\rm div} \left (a(|\nabla u(x)|)\nabla u(x)\right )& =& g(x,u), & \mbox{in} \Omega u& = &0, & \mbox{on} \partial\Omega, \end{array} \right .\)where \(\Omega \) is a bounded domain in \({\mathbb R}^N\), \(g\in C(\overline{\Omega}\times\mathbb R,\mathbb R)\), and the function \(\phi(s)= sa(|s|)\) is an increasing homeomorphism from \({\mathbb R}\) onto \({\mathbb R}\). Under appropriate conditions on \(\phi\), \(g\), and the Orlicz-Sobolev conjugate \(\Phi_*\) of \(\Phi(s)=\int_0^s\phi(t) dt,\) (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type.

Mathematics Subject Classification (1991): 35J20, 35J25 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Ph. Clément
    • 1
  • M. García-Huidobro
    • 2
  • R. Manásevich
    • 3
  • K. Schmitt
    • 4
  1. 1.Department of Pure Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands NL
  2. 2.Departamento de Matemáticas, P. Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile CL
  3. 3.Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile CL
  4. 4.Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA US

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