On critical logarithmic double phase problems with locally defined perturbation

Abstract

This paper deals with critical logarithmic double phase problems of the form$-\mathrm{div}K\left(u\right)=g(x,u)+|u{|}^{p^{⁎}-2}u\text{in }\Omega ,u=0\text{on }\partial \Omega ,$ where $\mathrm{div}K$ is the logarithmic double phase operator defined by$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u+\mu \left(x\right)(\log (e+\left|\nabla u\right|)+\frac{\left|\nabla u\right|}{q(e+|\nabla u\left|\right)})\left|\nabla u{|}^{q-2}\nabla u\right),$ e is Euler's number, $\Omega \subset R^N$, $N\ge 2$, is a bounded domain with Lipschitz boundary ∂Ω, $1<p<N$, $p<q<p^{⁎}=\frac{Np}{N-p}$, $0\le \mu (\cdot )\in L^{\infty }\left(\Omega \right)$ and $g:\Omega \times [-\xi ,\xi ]\to R$ for $\xi >0$ is a locally defined Carathéodory function satisfying a certain behavior near the origin. Based on appropriate truncation techniques and a suitable auxiliary problem, we prove the existence of a whole sequence of sign-changing solutions of the problem above which converges to 0 in the logarithmic Musielak-Orlicz Sobolev space $W_0^{1,H_{\log }}\left(\Omega \right)$ and in $L^{\infty }\left(\Omega \right)$.