Opuscula Math. 44, no. 4 (2024), 445-470
https://doi.org/10.7494/OpMath.2024.44.4.445
Opuscula Mathematica
Study of fractional semipositone problems on RN
Abstract. 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 \(r \in [\frac{2N}{N-2s}, \infty]\).
Keywords: semipositone problems, fractional operator, uniform regularity estimates, positive solutions.
Mathematics Subject Classification: 35R11, 35J50, 35B65, 35B09.
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- Nirjan Biswas
- Tata Institute of Fundamental Research, Centre For Applicable Mathematics, Post Bag No. 6503, Sharada Nagar, Bangalore 560065, India
- Communicated by J.I. Díaz.
- Received: 2024-01-01.
- Revised: 2024-02-27.
- Accepted: 2024-03-01.
- Published online: 2024-04-29.
