Normalized homoclinic solutions of discrete nonlocal double phase problems

Abstract

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian: ( − Δ𝔻)pαu(k) + μ(−Δ 𝔻)qβu(k) + ω(k)|u(k)|p−2u(k) = λ|u(k)|q−2u(k) + h(k)|u(k)|r−2u(k) for k ∈ ℤ,∑k∈ℤ|u(k)|q = ρq > 0, u(k) → 0 as |k|→∞, where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text]([Formula: see text] or [Formula: see text], [Formula: see text] or [Formula: see text]) is the discrete fractional [Formula: see text]-Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

Keywords

• Double phase problem
• discrete fractional -Laplacian
• homoclinic solutions
• normalized solutions