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Study of fractional semipositone problems on RN

Biswas Nirjan

Opuscula Mathematica

2024

Vol. 44, no. 4

445--470

Let s ∈ (0, 1) and N > 2s. In this paper, we consider the following class of nonlocal semipositone problems: (−Δ)su = g(x)ƒa(u) in RN, u > 0 in RN, where the weight g ∈ L1(RN) ∩ L∞(RN) is positive, a > 0 is a parameter, and ƒa ∈ C(R) is strictly negative on (−∞, 0]. For ƒa having subcritical growth and weaker Ambrosetti–Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution ua, provided a is near zero. To obtain the positivity of ua, we establish a Brezis–Kato type uniform estimate of (ua) in Lr(RN) for every r ∈ [formula].