Singularly perturbed critical Choquard equations

Abstract

In this paper we study the semiclassical limit for the singularly perturbed Choquard equationwhere 0 < µ < 3, ε is a positive parameter, V, Q are two continuous real function on R 3 and G is the primitive of g which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on g, we first establish the existence of ground states for the critical Choquard equation with constant coefficients. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods.2010 Mathematics Subject Classification. 35J20,35J60, 35B33.

Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains

“…Then Choquard applied it to model an electron trapped in its own hole, in an approximation to the Hartree-Fock theory of onecomponent plasma [14]. In some cases equation (1) is known also as the Schrödinger-Newton equation. There are a lot of studies about the existence, multiplicity and properties of solutions of the nonlinear Choquard equation (1).…”

“…In some cases equation (1) is known also as the Schrödinger-Newton equation. There are a lot of studies about the existence, multiplicity and properties of solutions of the nonlinear Choquard equation (1). In particular, if N = 3, q = 2, µ = 1, and the potential is constant, the existence of ground states of equation (1) was established in [14] and [15] through variational methods, while uniqueness and nondegeneracy were obtained respectively in [14] and [13,22].…”

“…There are a lot of studies about the existence, multiplicity and properties of solutions of the nonlinear Choquard equation (1). In particular, if N = 3, q = 2, µ = 1, and the potential is constant, the existence of ground states of equation (1) was established in [14] and [15] through variational methods, while uniqueness and nondegeneracy were obtained respectively in [14] and [13,22]. For general q and µ, regularity, positivity, radial symmetry and decay property of the ground states were shown in [7,17,19].…”

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

“…In addition, Alves and Yang [5] investigated the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Very recently, in an interesting paper, Alves et al [3] study (1.4) with a critical growth, they consider the critical problem with both linear potential and nonlinear potential, and showed the existence, multiplicity and concentration behavior of solutions when the linear potential has a global minimum or maximum. On the contrary, the results about fractional Choquard equation (1.1) are relatively few.…”

Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

“…In another research, 5 Gao and Yang investigated Equation 1 with upper critical exponent N+ N−2 in a bounded domain with Lipschitz boundary, and established the existence, multiplicity, and nonexistence of nontrivial solutions to the critical Choquard equation. For details and recent works, we refer to other literature [6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”