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On boundary value problems for a discrete generalized Emden–Fowler equation. (English) Zbl 1112.39011

The authors study the following second order self-adjoint difference equation \[ \Delta[p(n)\Delta u(n-1)]+q(n)u(n)=f(n,u(n)), \] with boundary value conditions \[ u(a)+\alpha u(a+1)=A, \quad u(b+2)+\beta u(b+1)=B. \] The above equation is the discrete analogue of the following self-adjoint differential equation \((p(t)y')'+q(t)y=f(t,y)\), which is a generalization of Emden-Fowler equation \(\frac{t}{dt}\left(t^\rho \frac{du}{dt}\right)+t^\delta u^\gamma=0.\) In this paper the critical point theory is well used to the above boundary value problem (BVP). The existence of solutions of the BVP is transferred into the existence of critical points of some functional. By the saddle point theorem, some results of existence are established for the sublinear and superlinear cases, respectively, as well as the Lipschitzian case.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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