TY  - JOUR
ID  - Biswas2024
LB  - Biswas2024
AU  - Biswas, Nirjan
TI  - Study of fractional semipositone problems on R<sup>N</sup>
ST  - Study of fractional semipositone problems on R<sup>N</sup>
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2024
VL  - 44
IS  - 4
SP  - 445
EP  - 470
DO  - https://doi.org/10.7494/OpMath.2024.44.4.445
UR  - https://doi.org/10.7494/OpMath.2024.44.4.445
AB  - Let \(s\in (0,1)\) and \(N\gt 2s\). In this paper, we consider the following class of nonlocal semipositone problems: \[(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,\] where the weight \(g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)\) is positive, \(a\gt 0\) 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 \(r \in [\frac{2N}{N-2s}, \infty]\).
KW  - semipositone problems
KW  - fractional operator
KW  - uniform regularity estimates
KW  - positive solutions
SN  - 1232-9274
ER  -