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On $p$-fractional weakly-coupled system with critical nonlinearities

Nirjan Biswas, Souptik Chakraborty

TL;DR

The paper analyzes a $p$-fractional weakly-coupled system with critical nonlinearities on $\mathbb{R}^d$ driven by the operator $(-\Delta_p)^s$ with $0<s<1$, $d>sp$, and $\alpha+\beta=p_s^*$. It develops a global compactness framework by providing a full Palais–Smale decomposition of the associated energy functional $I_{f,g}$, revealing how PS sequences can split into a weak limit solving the full system plus a finite number of homogeneous bubbles, with precise energy and mutual-orthogonality properties. Leveraging this decomposition, the authors prove the existence of a positive, negative-energy solution when $\ker(f)=\ker(g)$ (for a restricted range of $s$), and they derive conditions under which the two components are distinct. The results advance the theory of nonlocal coupled systems at critical growth and give a robust variational toolkit for non-homogeneous fractional $p$-Laplacian systems on $\mathbb{R}^d$.

Abstract

This paper deals with the following nonlocal system of equations: \begin{align}\tag{$\mathcal S$}\label{MAT1} (-Δ_p)^s u = \fracα{p_s^*}|u|^{α-2}u|v|^β+f(x) \text{ in } \mathbb{R}^{d}, \, (-Δ_p)^s v = \fracβ{p_s^*}|v|^{β-2}v|u|^α+g(x) \text{ in } \mathbb{R}^{d},\; u,v >0 \mbox{ in } \mathbb{R}^{d}, \end{align} where $0<s<1<p< \infty$, $d>sp$, $α,β>1$, $α+β=\frac{dp}{d-sp}$, and $f,g$ are nontrivial nonnegative functionals in the dual space of $\mathcal{D}^{s,p}(\mathbb{R}^{d})$. The primary objective of this paper is to present a global compactness result that offers a complete characterization of the Palais-Smale sequences of the energy functional associated with \eqref{MAT1}. Using this characterization, within a certain range of $s$, we establish the existence of a solution with negative energy for \eqref{MAT1} when $\ker(f)=\ker(g)$.

On $p$-fractional weakly-coupled system with critical nonlinearities

TL;DR

The paper analyzes a -fractional weakly-coupled system with critical nonlinearities on driven by the operator with , , and . It develops a global compactness framework by providing a full Palais–Smale decomposition of the associated energy functional , revealing how PS sequences can split into a weak limit solving the full system plus a finite number of homogeneous bubbles, with precise energy and mutual-orthogonality properties. Leveraging this decomposition, the authors prove the existence of a positive, negative-energy solution when (for a restricted range of ), and they derive conditions under which the two components are distinct. The results advance the theory of nonlocal coupled systems at critical growth and give a robust variational toolkit for non-homogeneous fractional -Laplacian systems on .

Abstract

This paper deals with the following nonlocal system of equations: \begin{align}\tag{}\label{MAT1} (-Δ_p)^s u = \fracα{p_s^*}|u|^{α-2}u|v|^β+f(x) \text{ in } \mathbb{R}^{d}, \, (-Δ_p)^s v = \fracβ{p_s^*}|v|^{β-2}v|u|^α+g(x) \text{ in } \mathbb{R}^{d},\; u,v >0 \mbox{ in } \mathbb{R}^{d}, \end{align} where , , , , and are nontrivial nonnegative functionals in the dual space of . The primary objective of this paper is to present a global compactness result that offers a complete characterization of the Palais-Smale sequences of the energy functional associated with \eqref{MAT1}. Using this characterization, within a certain range of , we establish the existence of a solution with negative energy for \eqref{MAT1} when .
Paper Structure (7 sections, 15 theorems, 164 equations)

This paper contains 7 sections, 15 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Global compactness results
  3. Technical lemmas
  4. Behavior of bounded sequences in under the action of group of isometries
  5. Refined embeddings of in the Morrey spaces
  6. The Palais-Smale decomposition
  7. Existence of a positive solution for non-homogeneous system

Key Result

Theorem 1.2

Let $s\in (0,1)$ and $p \in (1, \infty)$. Let $\alpha, \beta > 1,\, \alpha + \beta = p^*_s$. Assume that $f, g$ are nontrivial nonnegative functionals in the dual space of ${\mathcal{D}^{s,p}}$. Let $\{ (u_n, v_n ) \}$ be a (PS) sequence for $I_{f,g}$ at level $\eta$. Then there exists a subsequence such that where $o_n(1) \rightarrow 0$ in $\mathcal{W}$ and $C_{x_n^i, r_n^i}\tilde{u}_i(x)\colone

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2: (PS) decomposition for nonhomogeneous system
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 22 more