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.
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Received April 22, 1999 / Accepted June 11, 1999 / Published online April 6, 2000
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Clément, P., García-Huidobro, M., Manásevich, R. et al. Mountain pass type solutions for quasilinear elliptic equations. Calc Var 11, 33–62 (2000). https://doi.org/10.1007/s005260050002
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DOI: https://doi.org/10.1007/s005260050002