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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 $$\left\{\matrix \Delta(|\Delta u(k - 1)|^{p(k-1)-2} \Delta u(k - 1)) = \lambda f(k,u(k)), \\ u(0) = u(T + 1) = 0,\endmatrix\right.$$ where $\lambda > 0$ is a numerical parameter, $f:\ [0,T + 1] \times {\Bbb R}^{T+2} \to {\Bbb R}$, $p:\ [0,T + 1] \to {\Bbb R}_+$. The Euler functional corresponding to this problem is $$J_\lambda(u) = \sum_{k=1}^{T+1} \frac1{p(k - 1)} |\Delta u(k - 1)|^{p(k-1)} - \lambda \sum_{k=1}^T F(k,u(k)), \quad F(k,u) = \int_0^u f(k,t) \, dt.$$ 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.

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