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Interpolation inequality and some applications

Abdellaziz Harrabi

Abstract

We investigate {\bf explicit} universal estimate of finite Morse index solutions to polyharmonic equations. \,Differently to previous works \cite{BL2, DDF, fa, H1}, propose here a direct proof using a new interpolation inequality and a delicate boot-strap argument under large superlinear and subcritical growth conditions to show that the universal constant grows as a power function of the Morse index.\, Also, our interpolation inequality allows us to provide local $L^p$-$W^{2r,p}$ estimate.

Interpolation inequality and some applications

Abstract

We investigate {\bf explicit} universal estimate of finite Morse index solutions to polyharmonic equations. \,Differently to previous works \cite{BL2, DDF, fa, H1}, propose here a direct proof using a new interpolation inequality and a delicate boot-strap argument under large superlinear and subcritical growth conditions to show that the universal constant grows as a power function of the Morse index.\, Also, our interpolation inequality allows us to provide local - estimate.

Paper Structure

This paper contains 8 sections, 6 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Interpolation inequalities.
  3. Explicit universal estimate.
  4. Proofs of Lemmas \ref{['00']} and \ref{['t1']}.
  5. Proof of Lemma \ref{['t1']}.
  6. Proof of Theorem \ref{['univ']}.
  7. Preliminary results.
  8. End of the proof of Theorem \ref{['univ']}.

Key Result

Lemma 1.1

There exist $\psi \in C_c^\infty (\omega')$ and a positive constant $C$ depending only on $(n,p,k,m)$ such that Moreover, we have

Theorems & Definitions (7)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.1
  • Remark 1.1
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.1