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Higher-order evolution inequalities with Hardy potential on the Korányi ball

Mohamed Jleli, Michael Ruzhansky, Bessem Samet, Berikbol T. Torebek

Abstract

We consider a higher order in (time) semilinear evolution inequality posed on the Korányi ball under an inhomogeneous Dirichlet-type boundary condition. The problem involves an inverse-square potential $λ/|ξ|_\mathbb{H}^2$, where $λ\geq -(Q-2)^2/4$ and a general weight function $V$ depending on the space variable in front of the power nonlinearity. We first establish a general nonexistence result for the considered problem. Next, in the special case $V(ξ):=|ξ|_\mathbb{H}^a$, $a\in \mathbb{R}$, we prove the sharpness of our nonexistence result and show that the problem admits three different critical behaviors according to the value of the parameter $λ$.

Higher-order evolution inequalities with Hardy potential on the Korányi ball

Abstract

We consider a higher order in (time) semilinear evolution inequality posed on the Korányi ball under an inhomogeneous Dirichlet-type boundary condition. The problem involves an inverse-square potential , where and a general weight function depending on the space variable in front of the power nonlinearity. We first establish a general nonexistence result for the considered problem. Next, in the special case , , we prove the sharpness of our nonexistence result and show that the problem admits three different critical behaviors according to the value of the parameter .
Paper Structure (10 sections, 18 theorems, 140 equations)

This paper contains 10 sections, 18 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Auxiliary results
  5. Admissible test functions
  6. A priori estimate
  7. Estimates of $J_i(\varphi)$
  8. Proofs of the main results
  9. Proof of Theorem \ref{['T1']}
  10. Proof of Theorem \ref{['T2']}

Key Result

Theorem 3.3

Let $k\geq 1$ be an integer, $p>1$, $\lambda\geq -\left(\frac{Q-2}{2}\right)^2$, $V=V(\xi)>0$ a.e. in $B_\mathbb{H}$ and $V^{\frac{-1}{p-1}}\in L^1_{\mathop{\mathrm{loc}}\nolimits}(B_\mathbb{H}\backslash\{0\},K\psi\,d\xi)$. Assume that $f\in L^{1,+}(\partial B_\mathbb{H})$. If then P--BC admits no weak solution.

Theorems & Definitions (36)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Corollary 3.6
  • Definition 3.7: Critical exponent of the first kind
  • Definition 3.8: Critical exponent of the second kind
  • Definition 3.9: Critical exponent of the third kind
  • Corollary 3.10
  • ...and 26 more