NONTRIVIAL SOLUTIONS OF THE DIRICHLET PROBLEM FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS

Abstract

Under the mild conditions by using the theory of the generalized Sobolev space and minimax theory we get the existence of solutions for a wide class of nonlinear Dirichlet problem as follows {ddxi[Gi(x, Du)]+f(x, u)=0, x∈Ω,u|∂Ω=0,where Ω is a bounded domain in Rn,Gi(x,q)=∂G∂qi,q=(qi,q2,⋯, qn). This equation is the Euler equation of the functional I(u)=∫Ω[G(x, Du)-F(x, u)]dx,where F(x, s)=∫0sf(x, t)dt.