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A Detailed Analysis on Sharpened Singular Adams-Type Inequalities

Deepak Kumar Mahanta, Tuhina Mukherjee, Abhishek Sarkar

Abstract

We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.

A Detailed Analysis on Sharpened Singular Adams-Type Inequalities

Abstract

We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on . A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the -biharmonic operator with singular exponential growth.

Paper Structure

This paper contains 8 sections, 24 theorems, 289 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. Sharp singular Adams' type inequality: Proof of Theorem \ref{['thm1.1']}
  4. Sharp singular concentration-compactness principle: Proof of Theorem \ref{['thm1.2']}
  5. Asymptotic behavior of subcritical sharp singular Adams' type Inequality: Proof of Theorem \ref{['thm5.1']}
  6. Equivalence of critical and subcritical sharp singular Adams' type inequalities: Proof of Theorem \ref{['thm5.22']}
  7. The compact embedding result: Proof of Theorem \ref{['thm1.4']}
  8. Existence of nontrivial weak solutions: Proof of Theorem \ref{['thm1.5']}

Key Result

Theorem 1.1

Let $n\geq4$, $1<p<\frac{n}{2}$ and $0\leq \gamma<n$ be hold. Then for all $0\leq\alpha\leq \beta_{\gamma,n}:=(1-\frac{\gamma}{n})\beta(n,2)$ and $u\in E$, there holds where $\Phi_{\alpha,j_0}(\cdot)$ is defined in eq1.55 and $\beta(n,2)=n[(n-2)\omega^{\frac{2}{n}}_{n-1}]^{\frac{n}{n-2}}$ with $\omega_{n-1}$ stands for the measure of the unit sphere in $\mathbb{R}^n$. Further, the constant $\beta

Theorems & Definitions (43)

  • Theorem 1.1: Sharp Singular Adams' Type Inequality
  • Theorem 1.2: Sharp Singular Concentration-Compactness Principle
  • Theorem 1.3: Subcritical Sharp Singular Adams' Type Inequality
  • Theorem 1.4: Critical Sharp Singular Adams' Type Inequality
  • Theorem 1.5
  • Example 1.6
  • Definition 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 33 more