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

Hichem Chtioui, Hichem Hajaiej, Lovelesh Sharma

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.

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

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.

Paper Structure

This paper contains 2 sections, 13 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Proof of the existence results

Key Result

Theorem 1.1

adimurthi1991neumann Let $p=2$, $n\geq 3$, and let $\alpha(x)$ and $\beta(x)$ be two functions satisfying condition eq1.2. Assume that the following conditions hold: (g.c.) There exists $x_0$ in the interior of $\Gamma_1$ such that and, in a neighborhood of $x_0$, $\Omega$ lies on one side of the tangent space of $\Gamma_1$ at $x_0$. ($\beta$.c.) The function $\beta(x)$ satisfies one of the follo

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Theorem 1.3
  • Remark 3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 14 more