Speaker
Description
We investigate a class of nonlinear p-Kirchhoff-type elliptic problems involving the critical Sobolev exponent. Such problems arise in several branches of mathematical physics and engineering, where nonlocal effects play a significant role in the modeling of complex phenomena. The presence of the critical Sobolev exponent introduces substantial analytical difficulties due to the lack of compactness of the Sobolev embedding, while the Kirchhoff term contributes an additional nonlocal structure to the problem.
The main objective of this study is to establish the existence of weak solutions under suitable assumptions on the nonlinear terms and the parameters involved. Our analysis is based on variational methods and critical point theory. More precisely, the associated energy functional is defined in an appropriate Sobolev space, and its geometric properties are carefully examined. By combining compactness arguments with variational techniques, we derive existence results for nontrivial solutions.
The study takes into account the influence of the p-Laplacian operator as well as the nonlocal Kirchhoff coefficient, which significantly affects the qualitative behavior of solutions. Special attention is devoted to the treatment of the critical growth term, whose presence requires refined analytical tools to overcome compactness issues. The obtained results extend and complement several existing contributions in the literature related to nonlinear Kirchhoff-type equations with critical nonlinearities.
Our findings demonstrate that, despite the difficulties generated by the critical Sobolev exponent and the nonlocal character of the problem, the existence of weak solutions can still be guaranteed through an appropriate variational framework. The results contribute to a better understanding of nonlinear elliptic equations with critical growth and may serve as a basis for further investigations concerning multiplicity, regularity, and qualitative properties of solutions.
Moreover, the presented methodology can be extended to investigate related classes of nonlinear differential equations arising in mathematical physics and engineering applications in future studies successfully.