Study of fractional semipositone problems on ℝ^N

2023/08/02 by Biswas, Nirjan · 2 citations
#35B09 #35B65 #35J50 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.00954

Abstract

Let s ∈ (0,1) and N >2s. In this paper, we consider the following class of nonlocal semipositone problems: (-Δ)^s u= g(x)f_a(u) \text in ℝ^N, u gt; 0 in ℝ^N, where the weight g ∈ L¹(ℝ^N) ∩ L^∞(ℝ^N) is positive, a>0 is a parameter, and f_a ∈ C(ℝ) is strictly negative on (-∞,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(ℝ^N) for every r ∈ [(2N)/(N-2s), ∞].

Cited by

• Positive solutions to semipositone problems on Heisenberg group
• Mixed local-nonlocal equations with critical nonlinearity on ℝ^N: Non-existence, Existence, and Multiplicity of positive solutions

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