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On the discrete boundary value problem for anisotropic equation. (English) Zbl 1233.39004

The article deals with the following boundary value problem

Δ(|Δu(k-1)| p(k-1)-2 Δu(k-1))=λf(k,u(k)),u(0)=u(T+1)=0,

where λ>0 is a numerical parameter, f:[0,T+1]× T+2 , p:[0,T+1] + . The Euler functional corresponding to this problem is

J λ (u)= k=1 T+1 1 p(k-1)|Δu(k-1)| p(k-1) -λ k=1 T F(k,u(k)),F(k,u)= 0 u f(k,t)dt·

The authors formulate natural conditions under which the Euler functional is coercive or anticoercive and, as a result, have got the corresponding solvability theorems. Further, they use a modification of the mountain pass lemma and get theorems about the existence of two and three solutions. In the end of the article a simple example is considered.

MSC:
39A12Discrete version of topics in analysis
39A10Additive difference equations
34B15Nonlinear boundary value problems for ODE
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