Existence of nontrivial solution for elliptic systems involving the \(p(x)\)-Laplacian (Q2875472)

The authors, after introducing some some basic properties of the variable exponent Sobolev spaces \(W^{1,p(x)}(\Omega), \) where \(\Omega \subset \mathbb{R}^n \) is a bounded domain with smooth boundary, study the existence of a nontrivial weak solution of suitable systems involving the nonhomogeneous \(p(x)\) Laplace operator.NEWLINENEWLINE The \(p(x)\) Laplace operator possesses more complicated nonlinearity than the \(p\) Laplace operator due to the fact that \(-\Delta_{p(x)}\) is not homogeneous, so in these kind of problems some special techniques will be needed.NEWLINENEWLINE The key argument in the proof is related to some suitable variational method and Ekeland's variational principle. The authors find out the extent to which well-posedness results for the Poisson problem with a Dirichlet boundary condition hold in the setting of weighted Sobolev spaces. They study the extent to which it is possible to depart from the basic case and consider \(L^p\) based Sobolev spaces with \(p\) not necessarily equal to 2.NEWLINENEWLINENEWLINEOn the geometric side, the main novelty is the fact that they succeed in formulating the main well-posedness results in the rather general setting of Lipschitz manifolds.NEWLINENEWLINE The results are sharp, by means of counterexamples, for a multitude of perspectives.NEWLINENEWLINENEWLINEIn this paper, the authors continue the study carried out in the paper [\textit{L. R. Duduchava} et al., Math. Nachr. 279, No. 9-10, 996--1023 (2006; Zbl 1112.58020)].NEWLINENEWLINENEWLINEBrewster and Mitrea considered weighted Sobolev spaces of arbitrary smoothness in Euclidean Lipschitz domains and proved that Stein's extension operator continues to work in this setting.NEWLINENEWLINENEWLINEMoreover, this is used to establish a very useful interpolation result.NEWLINENEWLINENEWLINELater, the authors study the trace theorem for such weighted Sobolev spaces and construct a boundary extension operator which serves as an inverse from the right for the trace mapping.NEWLINENEWLINENEWLINEBoundary value problems for elliptic systems with bounded measurable coefficients in Euclidean Lipschitz domains are also treated. Furthermore, these results are generalized to the setting of compact Lipschitz manifolds with boundary.