Abstract: Let Ω⊂Rⁿ be a bounded domain with a smooth boundary ∂Ω, L a strictly elliptic operator and c(x) ≥ 0 in Ω. In this paper we are concerned with the following Dirichlet problem
Lu=-Σ_i,j=1ⁿ∂/∂x_i(α_ij(x)∂u/∂x_j)+c(x)u=p(x,u), x∈Ω;u=0,x∈∂Ω (1)
with the growth condition (P₁):lim_|t|→∞|p(x,t)/(t)=0,where φ(t)=|t|^(n+2)/(n-2) for n > 2;=e|^t|α-|t|^α-1,1 < α < 2, for n=2. It is proved that if p(x, t) has all derivatives up to order l which are locally Hölder continuous in Ω×R. and if α_ij(x) ∈C_l+1,α(Ω) and c(x)∈C_l,α(Ω), then any weak solution in W₀^1,2 of (1) lies in C_l+2,α(Ω). Moreover, under the growth condition (P₁) and some additional assumptions, the existence of nontrivial solution of (1) is proved. The main difficulity here is that the simple bootstrapping procedure fails to apply directly to the case of the growth condition (P₁).