On logarithmic double phase problems

Abstract

In this paper we introduce a new logarithmic double phase type operator of the form$\begin{array}{cc}Gu& :=-\mathrm{div}({\left|\nabla u\right|}^{p\left(x\right)-2}\nabla u+\mu (x\left)[\log (e+\left|\nabla u\right|)+\frac{\left|\nabla u\right|}{q\left(x\right)(e+\left|\nabla u\right|)}]{\left|\nabla u\right|}^{q\left(x\right)-2}\nabla u\right),\end{array}$ whose energy functional is given by$u\mapsto \underset{\Omega }{\int }(\frac{{\left|\nabla u\right|}^{p\left(x\right)}}{p\left(x\right)}+\mu (x)\frac{{\left|\nabla u\right|}^{q\left(x\right)}}{q\left(x\right)}\log (e+\left|\nabla u\right|\left)\right)d\text{x},$ where $\Omega \subseteq R^N$, $N\ge 2$, is a bounded domain with Lipschitz boundary ∂Ω, $p,q\in C\left(\bar{\Omega }\right)$ with $1<p\left(x\right)\le q\left(x\right)$ for all $x\in \bar{\Omega }$ and $0\le \mu (\cdot )\in L^1\left(\Omega \right)$. First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces $W^{1,H_{\log }}\left(\Omega \right)$ and $W_0^{1,H_{\log }}\left(\Omega \right)$ with $H_{\log }(x,t)=t^{p\left(x\right)}+\mu \left(x\right)t^{q\left(x\right)}\log (e+t)$ for $(x,t)\in \bar{\Omega }\times [0,\infty )$ are separable, reflexive Banach spaces and $W_0^{1,H_{\log }}\left(\Omega \right)$ can be equipped with the equivalent norm$\inf \{\lambda >0:\underset{\Omega }{\int }[{\left|\frac{\nabla u}{\lambda }\right|}^{p\left(x\right)}+\mu (x){\left|\frac{\nabla u}{\lambda }\right|}^{q\left(x\right)}\log (e+\frac{\left|\nabla u\right|}{\lambda })]d\text{x}\le 1\}.$ We also prove several embedding results for these spaces and the closedness of $W^{1,H_{\log }}\left(\Omega \right)$ and $W_0^{1,H_{\log }}\left(\Omega \right)$ under truncations. In addition we show the density of smooth functions in $W^{1,H_{\log }}\left(\Omega \right)$ 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. 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$Gu=f(x,u)\text{in }\Omega ,u=0\text{on }\partial \Omega$ 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 Poincaré-Miranda existence theorem.