Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity

Abstract

We consider the following Kirchhoff -Choquard equationwhere Ω is a bounded domain in R N (N ≥ 3) with C 2 boundary, 2 * µ = 2N −µ N −2 , 1 < q ≤ 2, and f is a continuous real valued sign changing function. When 1 < q < 2, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also prove the existence of a positive solution when q = 2 using the Mountain Pass Lemma.

Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

“…For fractional Kirchho problems with singular nonlinearity, we also refer the interested readers to [44]. Very recently, Goel and Sreenadh [16] used the similar method as in [12] to discuss the following critical Choquard-Kirchho type problem…”

Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

“…The mathematical aspects of the model (1) in the absence of memory, that is, b = 0, have been widely investigated theoretically as well as numerically by several authors. The Kirchhoff-type nonlocal problems attained a lot of interest by elliptic PDE community, for instance, nonlocal perturbations of stationary Kirchhoff problems are studied in Goel and Sreenadh 11 and Sing. 12 The analysis of water wave models with nonlocal terms is studied in Goubet and Manoubi 13 and magnetic field equations of integro-Kirchhoff equations in Mingqi et al 14 Finally, we also cite Mingqi et al 15 and Arora et al 16 for problems involving critical growth nonlocal terms motivated by noncompact embeddings in the limiting dimensions.…”

Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents

“…The problem (1.1) is a general version of a model without Hartree term, which was firstly proposed in 1883 by Kirchhoff. Recently, Goel and Sreenadh established a steady-state Kirchhoff-Choquard model in [13]. For the related problems about this topic, we refer to [4,17] for latest literature.…”

“…As the equation involves convex-concave nonlinearities, they took advantage of Nehari manifold and minimization approach to get multiple results. If f is a sign-changing weight function, replace a + ε Ω |∇u| 2 with a + ε p ( Ω |∇u| 2 ) θ−1 in (1.1), Goel et al [13] used Nehari manifold and concentration compactness principle to obtain positive solutions to the Kirchhoff-Choquard problem with critical exponential growth nonlinearity. Xiang, Rȃdulescu and Zhang in [28] studied a critical fractional Kirchhoff-Choquard problem with magnetic field and established the existence of nontrivial radial solutions in non-degenerate and degenerate cases.…”