On $p$-fractional weakly-coupled system with critical nonlinearities
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2025
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| author | Biswas, Nirjan Chakraborty, Souptik |
| author_facet | Biswas, Nirjan Chakraborty, Souptik |
| contents | 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 $\ker(f)=\ker(g)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_04994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $p$-fractional weakly-coupled system with critical nonlinearities Biswas, Nirjan Chakraborty, Souptik Analysis of PDEs Functional Analysis 35R11, 35A15, 35B33, 35J60 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 $\ker(f)=\ker(g)$. |
| title | On $p$-fractional weakly-coupled system with critical nonlinearities |
| topic | Analysis of PDEs Functional Analysis 35R11, 35A15, 35B33, 35J60 |
| url | https://arxiv.org/abs/2501.04994 |