On logarithmic double phase problems

In this paper we introduce a new logarithmic double phase type operator of the form Gu:=-div(|del u|(p(x)-2)del u+mu(x)[log(e+|del u|)+|del u|/ q(x)(e+|del u|)]|del u|(q(x)-2)del u), whose energy functional is given by u integral(Omega)(|del u|(p(x))/ p(x)+mu(x)|del u|(q(x))/ q(x) log(e+|del u|))dx, where Omega subset of RN, N >= 2, is a bounded domain with Lipschitz boundary partial derivative Omega, p,q is an element of C(Omega) with 1<p(x)<= q(x) for all x is an element of Omega and 0 <=mu(& sdot;)is an element of L1(Omega). First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces W1,Hlog(Omega) and W01,Hlog(Omega) with Hlog(x,t)=tp(x)+mu(x)tq(x)log(e+t) for (x,t)is an element of Omega x[0,infinity) are separable, reflexive Banach spaces and W01,Hlog(Omega) can be equipped with the equivalent norm inf { lambda>0:integral(Omega)[|del u/ lambda|(p(x))+ mu(x) |del u/ lambda|(q(x)) log(e+|del u|/ lambda)] dx <= 1}. We also prove several embedding results for these spaces and the closedness of W-1,W-Hlog(Omega) and W-0(1,Hlog)(Omega) under truncations. In addition we show the density of smooth functions in W-1,W-Hlog(Omega) 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 C-1. 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 Omega, u=0 on partial derivative Omega 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 Poincare-Miranda existence theorem. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). MSC: 35A01; 35J20; 35J25; 35J62; 35J92; 35Q74