Critical growth elliptic problems with Choquard type nonlinearity:A survey

Abstract

This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.

Regularity results on a class of doubly nonlocal problems

“…But in case of nonlocal nonlinearity, in particular, Choquard equation, a particular case to our result had been answered by [16] for the Laplacian operator. For the general nonlinearity, this issue is recently posed as an open problem in [23]. In this article, we also provide a full answer to this open problem.…”

“…where Ω ⊂ R N , N ≥ 3 is a bounded domain having smooth boundary ∂Ω, λ > 0, 0 < µ < N and 2 * µ = 2N −µ N −2 . Later, many researchers studied the Choquard equation for the existence and multiplicity of solutions, for instance see [4,15,23] and references therein.…”

“…The nonlocal equations with Hartree-type nonlinearities were used to model the dynamics of pseudo-relativistic boson stars. In fractional quantum mechanics, fractional Schrödinger equations play an important role, for instance see [13,37,23]. For the existence and multiplicity results on fractional Laplacian, readers can refer to [24] and references therein.…”

Adams–Moser–Trudinger inequality in the Cartesian product of Sobolev spaces and its applications

“…These kind of non-linearities also play a crucial role in the Bose-Einstein condensation [8]. For interested readers, we refer the recent survey paper on Choquard equations by Moroz and Schaftingen [21] and Mukherjee and Sreenadh [22]. In 2014, Lü [17] studied the non-degenerate Choquard equation with Kirchhoff operator in R 3 and using the method of Nehari manifold established the existence of ground state solution.…”

Brezis-Nirenberg type result for Kohn Laplacian with critical Choquard Nonlinearity

“…For more details on recent works on Choquard equation we refer to [30,31] and the reference therein.…”