On logarithmic double phase problems

Abstract

In this paper we introduce a new logarithmic double phase type operator of the form [Formula presented] whose energy functional is given by [Formula presented] where Ω⊆RN, N≥2, is a bounded domain with Lipschitz boundary ∂Ω, p,q∈C(Ω‾) with 1<p(x)≤q(x) for all x∈Ω‾ and 0≤μ(⋅)∈L1(Ω). First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces W1,H(Ω) and W01,H(Ω) with Hlog(x,t)=tp(x)+μ(x)tq(x)log⁡(e+t) for (x,t)∈Ω‾×[0,∞) are separable, reflexive Banach spaces and W01,H(Ω) can be equipped with the equivalent norm [Formula presented] We also prove several embedding results for these spaces and the closedness of W1,H(Ω) and W01,H(Ω) under truncations. In addition we show the density of smooth functions in W1,H(Ω) even in the case of an unbounded domain by supposing Nekvinda's decay condition on p(⋅). The second part is devoted to the properties of the operator and it turns out that it is bounded, continuous, strictly monotone, of type (S+), coercive and a homeomorphism. Also, the related energy functional is of class C1. As a result of independent interest we also present a new version of Young's inequality for the product of a power-law and a logarithm. In the last part of this work we consider equations of the form Gu=f(x,u)in Ω,u=0on ∂Ω with superlinear right-hand sides. We prove multiplicity results for this type of equation, in particular about sign-changing solutions, by making use of a suitable variation of the corresponding Nehari manifold together with the quantitative deformation lemma and the Poincaré-Miranda existence theorem. © 2025 The Author(s)