Logarithmic double phase problems with critical growth on the boundary

Abstract In this paper, we study logarithmic double phase problems with critical growth on the boundary of the form $$\begin{aligned} -\operatorname {div} {\mathcal {L}}(u)=-|u|^{p-2}u \quad \text {in } \Omega , \quad {\mathcal {L}}(u)\cdot \nu = f(x,u)+ |u|^{p_*-2}u \quad \text {on } \partial \Omega , \end{aligned}$$ - div L ( u ) = - | u | p - 2 u in Ω , L ( u ) · ν = f ( x , u ) + | u | p ∗ - 2 u on ∂ Ω , where $$\operatorname {div} {\mathcal {L}}$$ div L stands for the logarithmic double phase operator given by $$\begin{aligned} \operatorname {div} \left( |\nabla u|^{p-2} \nabla u + \mu (x) \left[ \log (e + |\nabla u|) + \frac{|\nabla u|}{q(e + |\nabla u|)} \right] |\nabla u|^{q-2} \nabla u \right) , \end{aligned}$$ div | ∇ u | p - 2 ∇ u + μ ( x ) log ( e + | ∇ u | ) + | ∇ u | q ( e + | ∇ u | ) | ∇ u | q - 2 ∇ u , e is Euler’s number, $$\nu (x)$$ ν ( x ) is the outer unit normal of $$\Omega $$ Ω at $$x \in \partial \Omega $$ x ∈ ∂ Ω , $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N , $$N \ge 2$$ N ≥ 2 , is a bounded domain with Lipschitz boundary $$\partial \Omega $$ ∂ Ω , $$1< p < N$$ 1 < p < N , $$p< q < p_* = \frac{(N - 1)p}{N - p}$$ p < q < p ∗ = ( N - 1 ) p N - p , $$\mu \in L^\infty (\Omega )$$ μ ∈ L ∞ ( Ω ) with $$\mu \ge 0$$ μ ≥ 0 , and $$f :\partial \Omega \times [-{\mathcal {K}}, {\mathcal {K}}] \rightarrow {\mathbb {R}}$$ f : ∂ Ω × [ - K , K ] → R for some $${\mathcal {K}} > 0$$ K > 0 is a Carathéodory function, just locally defined with a specific behavior near the origin. Using suitable truncation methods and an appropriate auxiliary problem along with an equivalent norm in our function space, we establish the existence of an entire sequence of sign-changing solutions to the above problem, which converges to zero in both the logarithmic Musielak-Orlicz Sobolev space $$W^{1, {\mathcal {H}}_{\log }}(\Omega )$$ W 1 , H log ( Ω ) and in $$L^{\infty }(\Omega )$$ L ∞ ( Ω ) .