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Positive homoclinic solutions for the discrete $$p$$-Laplacian with a coercive weight function. (English) Zbl 1313.39004
By using a mountain-pass theorem due to P. Pucci and J. Serrin [J. Differ. Equations 60, 142–149 (1985; Zbl 0585.58006)], the authors prove the existence of positive solutions for the nonlinear second-order difference equation $\begin{cases} -\Delta \phi _p(\Delta u(k-1))+a(k)\phi _p(u(k))=f(k,u(k)),\,\,\,k\in \mathbb {Z},\\ u(k)\to 0,\,\,\,\text{as}\,\,\,| k| \to \infty .\end{cases}$ Here $$p\in \mathbb {R}$$, $$p>1$$, $$\phi _p(t)=| t| ^{p-2}t$$ for all $$t\in \mathbb {R}$$, $$\Delta u$$ is the forward difference operator defined by $$\Delta u(k-1)=u(k)-u(k-1)$$ for all $$k\in \mathbb {Z}$$, $$a:\mathbb {Z}\to \mathbb {R}$$ is a positive and coercive function, and $$f:\mathbb {Z}\times \mathbb {R}\to \mathbb {R}$$ is a continuous function which satisfies some additional assumptions.

##### MSC:
 39A10 Additive difference equations 47J30 Variational methods involving nonlinear operators 39A22 Growth, boundedness, comparison of solutions to difference equations
Zbl 0585.58006