## Homoclinic solutions of discrete $$\phi$$-Laplacian equations with mixed nonlinearities.(English)Zbl 1396.39008

Summary: By using critical point theory, we obtain some new sufficient conditions on the existence of homoclinic solutions of a class of nonlinear discrete $$\phi$$-Laplacian equations with mixed nonlinearities for the potentials being periodic or being unbounded, respectively. And we prove it is also necessary in some special cases. In addition, multiplicity results of homoclinic solutions for nonlinear discrete $$\phi$$-Laplacian equations with unbounded potentials have also been considered. In our paper, the nonlinearities can be mixed super $$p$$-linear with asymptotically $$p$$-linear at $$\infty$$ for $$p\geq 1$$. To the best of our knowledge, there is no such result for the existence of homoclinic solutions with discrete $$\phi$$-Laplacian before. Finally, an extension has also been considered.

### MSC:

 39A14 Partial difference equations 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 39A12 Discrete version of topics in analysis
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### References:

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