ISSN:
1432-0835
Keywords:
Mathematics Subject Classification (1991): 35J20, 35J25
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s005260050002
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