The Existence of Ground State Solutions for p-Laplacian Equations on Lattice Graphs

Abstract

In this paper, we study the nonlinear p-Laplacian equation

$-\Delta_{p}u+V(x)|u|^{p-2}u=f(x,u)$

on the lattice graph ℤ^N, where Δ_p is the discrete p-Laplacian, p ∈ (1, ∈). We study two cases of potential V. The first is periodic and the second tends to a constant at infinity, both of which are positive and bounded. The nonlinearity f is superlinear and satisfies the growth condition ∣f(x, u)∣ ≤ a(1 + ∣u∣^q−1) for some q >p. We first prove the equivalence of three function spaces on ℤ^N, which is quite different from the continuous case and allows us to remove the restriction q > p* in [Handbook of Nonconvex Analysis and Applications, 597–632, Somerville, MA: Int. Press, 2010], where p* is the Sobolev critical exponent. Then, using the method of Nehari ([Trans. Amer. Math. Soc., 1960, 95: 101–123] and [Acta Math., 1961, 105: 141–175]), we prove the existence of ground state solutions to the above equation.

Keywords

p-Laplacian / Nehari method / lattice graphs / 35Q55 / 39A14 / 58E30

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Peking University