Fractional Choquard equation with critical nonlinearities

Abstract

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacianwhere Ω is a bounded domain in R n with Lipschitz boundary, λ is a real parameter, s ∈ (0, 1), n > 2s and 2 * µ,s = (2n − µ)/(n − 2s) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.

Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

“…Chen and Liu [13] studied (1.7) with nonconstant linear potential and proved the existence of ground states without any symmetry property. For critical problem, Wang and Xiang [39] obtain the existence of infinitely many nontrivial solutions and the Brezis-Nirenberg type results can be founded in [33]. For other existence results we refer to [8,9,19,20,27,40,46] and the references therein.…”

Unbalanced (p,2)-fractional problems with critical growth

“…In this work authors proved regularity, existence, non-existence, symmetry as well as decay properties of solution. Mukherjee and Sreenadh [29] obtained Brezis-Nirenberg type results for problem involving fractional Laplacian with Choquard type nonlinearity having critical growth. In [36], Pucci et al studied problem involving critical Choquard nonlinearity with fractional p-Laplacian.…”

Variable order nonlocal Choquard problem with variable exponents

“…Here the author have considered −∆u = Ω |u| 2 * µ |x − y| µ(x,y) dy u 2 * µ −1 + λu, x ∈ Ω, and u = 0, x ∈ ∂Ω, where 2 * µ = (2N −µ) (N −2) , 0 < µ < N . For more results on Choquard problem involving concaveconvex nonlinearities we refer ( [30]- [31], [35]- [36]). In recent years, problems involving nonlocal operators have gained a lot of attentions due to their occurrence in real-world applications, such as, the thin obstacle problem, optimization, finance, phase transitions and also in pure mathematical research, such as, minimal surfaces, conservation laws etc.…”

“…In continuation to this, the problems involving quasilinear nonlocal fractional p-Laplace operator are extensively studied by many researchers including Squassina, Palatucci, Mosconi, Rădulescu et al (see [14,22,28,29] ), where the authors studied various aspects, viz., existence, multiplicity and regularity of the solutions of the quasilinear nonlocal problem involving fractional p-Laplace operator. Recently, Choquard problem involving nonlocal operators have been studied by Squassina et.al in [10] and Mukerjee and Sreenadh in [31]. In [31], authors have discussed the Brezis-Nirenberg type problem of nonlinear Choquard equation involving the fractional Laplacian As the variable growth on the exponent p in the local p(x)-Laplace operator, defined as div(|∇u| p(x)−2 ∇u), makes it more suitable for modeling the problems like image restoration, obstacle problems compared to p-Laplace operator, henceforth, it is a natural inquisitiveness to substitute the nonlocal fractional p-Laplace operator with the nonlocal operator involving variable exponents and variable order as defined in (1.2) and expect better modeling.…”

“…Recently, Choquard problem involving nonlocal operators have been studied by Squassina et.al in [10] and Mukerjee and Sreenadh in [31]. In [31], authors have discussed the Brezis-Nirenberg type problem of nonlinear Choquard equation involving the fractional Laplacian As the variable growth on the exponent p in the local p(x)-Laplace operator, defined as div(|∇u| p(x)−2 ∇u), makes it more suitable for modeling the problems like image restoration, obstacle problems compared to p-Laplace operator, henceforth, it is a natural inquisitiveness to substitute the nonlocal fractional p-Laplace operator with the nonlocal operator involving variable exponents and variable order as defined in (1.2) and expect better modeling. In analogy to the Lebesgue spaces with variable exponents (see [13,11]), recently Kaufmann et al introduced the fractional Sobolev spaces with variable exponents in [19].…”