Regularity for nonuniformly elliptic equations with \(p, q\)-growth and explicit \(x, u\)-dependence (Q6583718)

The paper under review studies the regularity of solutions to a class of nonuniformly elliptic equations under \(p, q\)-growth assumptions. The authors aim to address the higher regularity properties of solutions that explicitly depend on both the spatial variable \(x\) and the solution \(u\), while also imposing growth and integrability assumptions on lower-order terms. This explicit dependence adds significant complexity to the analysis, requiring advanced mathematical tools and careful treatment. Motivated by examples, the authors highlight the relevance of \(p, q\)-growth conditions in the context of regularity theory for elliptic equations. They show that generalized solutions achieve significant regularity properties, thereby advancing the theoretical understanding of these equations. This is particularly relevant in mathematical modeling and applications, where such growth conditions naturally arise in describing complex physical phenomena. Their results are obtained through clever use of sophisticated regularity techniques, such as Sobolev embeddings and Caccioppoli inequalities, providing a rigorous framework for tackling similar problems. The methods used are versatile and could inspire further studies in related fields, including the treatment of time-dependent or more general nonlinear equations.