On logarithmic double phase problems
Rakesh Arora, Ángel Crespo-Blanco, Patrick Winkert
TL;DR
The paper introduces a logarithmic double-phase operator $\mathcal{G}$ acting on a novel logarithmic Musielak–Orlicz Sobolev space $W^{1,\mathcal{H}_{\log}}(\Omega)$, addressing nonuniform ellipticity governed by variable exponents $p(x)$ and $q(x)$ with a modulating coefficient $\mu(x)$. It builds the underlying functional-analytic framework, proving separability, reflexivity, density of smooth functions, and robust embedding properties, and then establishes that the associated energy functional $I$ is $C^1$ with gradient operator $A$ that is bounded, continuous, strictly monotone and of type $(S_+)$. Using these tools, the authors obtain three nontrivial weak solutions to problems driven by $\mathcal{G}$ with superlinear nonlinearities: two constant-sign solutions and a sign-changing one, via a tailored Nehari-manifold approach, a quantitative deformation lemma, and the Poincaré–Miranda theorem. They also provide nodal-domain results and extend density results to unbounded domains under Nekvinda’s decay condition. The work introduces a new Young-type inequality adapted to products of power-laws and logarithms, and opens avenues for regularity and multiplicity analyses in logarithmic, nonuniformly elliptic variational problems.
Abstract
In this paper we introduce a new logarithmic double phase type operator of the form\begin{align*}\mathcal{G}u:=-\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u+μ(x)\left[\log(e+|\nabla u|)+\frac{|\nabla u|}{q(x)(e+|\nabla u|)}\right]|\nabla u|^{q(x)-2} \nabla u \right),\end{align*}where $Ω\subseteq\mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partialΩ$, $p,q\in C(\overlineΩ)$ with $1<p(x)\leq q(x)$ for all $x\in\overlineΩ$ and $0\leqμ(\cdot)\in L^1(Ω)$. First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces $W^{1,\mathcal{H}_{\log}}(Ω)$ and $W^{1, \mathcal{H}_{\log}}_0(Ω)$ with $\mathcal{H}_{\log}(x,t)=t^{p(x)}+μ(x)t^{q(x)}\log(e+t)$ for $(x,t)\in \overlineΩ\times [0,\infty)$ are separable, reflexive Banach spaces and $W^{1,\mathcal{H}_{\log}}_0(Ω)$ can be equipped with an equivalent norm. We also prove several embedding results for these spaces and the closedness of these spaces under truncations. In addition we show the density of smooth functions in $W^{1,\mathcal{H}_{\log}}(Ω)$ even in the case of an unbounded domain by supposing Nekvinda's decay condition on $p(\cdot)$. 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. 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 driven by our new operator 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.