Mixed local-nonlocal equations with critical nonlinearity on $\mathbb{R}^N$: Non-existence, Existence, and Multiplicity of positive solutions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913158576209920 |
|---|---|
| author | Biswas, Nirjan Chakraborty, Souptik Das, Paramananda |
| author_facet | Biswas, Nirjan Chakraborty, Souptik Das, Paramananda |
| contents | We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract_1}\tag{$\mathcal{P}_p$}
-Δ_p u+(-Δ_p)^s u=|u|^{p^*-2}u+f(x)\text{ in }\mathbb{R}^N, \end{equation} where $s \in (0,1), p \in (1, \infty), N>p$, $p^*=\frac{Np}{N-p}$, and $f$ is a nonnegative functional in the dual space of the ambient solution space. If $f \equiv0$, then we show that \eqref{main_prob_abstract_1} does not admit any nontrivial weak solution. This phenomenon stands in contrast to the purely local and purely nonlocal cases. On the other hand, if $f$ is a nontrivial nonnegative functional, we establish the existence of a positive weak solution to \eqref{main_prob_abstract_1} provided $\|f\|$ is small. For this purpose we prove the concentration compactness principle for the mixed operator $-Δ_p +(-Δ_p)^s$ in $\mathbb{R}^N$. We also discuss the multiplicity of positive weak solutions to \eqref{main_prob_abstract_1}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mixed local-nonlocal equations with critical nonlinearity on $\mathbb{R}^N$: Non-existence, Existence, and Multiplicity of positive solutions Biswas, Nirjan Chakraborty, Souptik Das, Paramananda Analysis of PDEs 35B09, 35B33, 35J20, 35J60 We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract_1}\tag{$\mathcal{P}_p$} -Δ_p u+(-Δ_p)^s u=|u|^{p^*-2}u+f(x)\text{ in }\mathbb{R}^N, \end{equation} where $s \in (0,1), p \in (1, \infty), N>p$, $p^*=\frac{Np}{N-p}$, and $f$ is a nonnegative functional in the dual space of the ambient solution space. If $f \equiv0$, then we show that \eqref{main_prob_abstract_1} does not admit any nontrivial weak solution. This phenomenon stands in contrast to the purely local and purely nonlocal cases. On the other hand, if $f$ is a nontrivial nonnegative functional, we establish the existence of a positive weak solution to \eqref{main_prob_abstract_1} provided $\|f\|$ is small. For this purpose we prove the concentration compactness principle for the mixed operator $-Δ_p +(-Δ_p)^s$ in $\mathbb{R}^N$. We also discuss the multiplicity of positive weak solutions to \eqref{main_prob_abstract_1}. |
| title | Mixed local-nonlocal equations with critical nonlinearity on $\mathbb{R}^N$: Non-existence, Existence, and Multiplicity of positive solutions |
| topic | Analysis of PDEs 35B09, 35B33, 35J20, 35J60 |
| url | https://arxiv.org/abs/2510.15507 |