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Double phase implicit obstacle problems with convection term and multivalued operator
Preview AbstractThis paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke’s generalized gradient. We develop a general framework to deliver an existence result for the double phase implicit obstacle problem under consideration. Our proof is based on the Kakutani–Ky Fan fixed point theorem together with the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded pseudomonotone mapping.
Normalized homoclinic solutions of discrete nonlocal double phase problems
Preview AbstractThe 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)αpu(k)+μ(−Δ𝔻)βqu(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)→0as |k|→∞,where α,β∈(0,1), ω:ℤ→(0,∞), 1<p≤q<∞, λ,μ∈ℝ, h∈ℓqq−r(ℤ) if 1<r<q, h∈ℓ∞(ℤ) if r>q, and (−Δ𝔻)sκ(s=α or β, κ=p or q) is the discrete fractional κ-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.