Opuscula Math. 44, no. 4 (2024), 445-470

Opuscula Mathematica

Study of fractional semipositone problems on RN

Nirjan Biswas

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.
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Cite this article as:
Nirjan Biswas, Study of fractional semipositone problems on RN, Opuscula Math. 44, no. 4 (2024), 445-470, https://doi.org/10.7494/OpMath.2024.44.4.445

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