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ON THE REGULARITY AND EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS OF SECOND ORDER
邓耀华
数学物理学报(英文版). 1987 (2):
217-228.
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=-Σi,j=1n∂/∂xi(αij(x)∂u/∂xj)+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 αij(x) ∈Cl+1,α(Ω) and c(x)∈Cl,α(Ω), then any weak solution in W01,2 of (1) lies in Cl+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).
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