Study of fractional semipositone problems on $\mathbb{R}^N$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913868932972544 |
|---|---|
| author | Biswas, Nirjan |
| author_facet | Biswas, Nirjan |
| contents | Let $s \in (0,1)$ and $N >2s$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*}
(-Δ)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00954 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Study of fractional semipositone problems on $\mathbb{R}^N$ Biswas, Nirjan Analysis of PDEs 35R11, 35J50, 35B65, 35B09 Let $s \in (0,1)$ and $N >2s$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-Δ)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$. |
| title | Study of fractional semipositone problems on $\mathbb{R}^N$ |
| topic | Analysis of PDEs 35R11, 35J50, 35B65, 35B09 |
| url | https://arxiv.org/abs/2308.00954 |