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
d/dx_i[G_i(x,Du)]+f(x,u)=0, x∈Q, u|_∂Q=0,
where Q is a bounded domain in Rⁿ,G_i(x,q)=∂G/∂q_i, q=(q₁,q₂,…,q_n). This equation is the Euler equation of the functional
I(u)=∫_Q[G(x,Du)-F(x,u)]dx,
where
F(x,s)=∫₀^xf(x,t)dt.