The role of the mean curvature in nonlinear p-Laplacian problems with critical exponent

TL;DR

This paper investigates the existence of least energy solutions for critical nonlinear p-Laplacian problems with mixed boundary conditions, highlighting differences from the linear case and challenges for different p values.

Contribution

It establishes existence results for the p-Laplacian with critical exponent, revealing fundamental differences from the linear case and between p<2 and p>2 scenarios.

Findings

Existence of least energy solutions proven for critical p-Laplacian problems.

Different boundary geometry effects dominate depending on whether p > 2 or p < 2.

Results differ significantly from the classical Laplacian case.

Abstract

We deal with critical nonlinear problems involving the p-Laplacian operator on bounded domains with mixed boundary conditions. We prove the existence of least energy solutions. Our work shows a significant difference between the semi-linear case p = 2 and the quasilinear case for the existence results. Moreover, neither the results for the Laplacian can be extended to the p-Laplacian, nor the method for the p-Laplacian can apply to the Laplacian setting. Additionally, the cases (p < 2 and p > 2) present different challenges and need to be studied separately. More precisely, when p > 2, the effect of the geometry of the boundary conditions dominates that one of the potential, whereas for p < 2 the opposite behavior holds true.