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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
Citations:
Zbl 0585.58006
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