Ground States for Mass Critical Two Coupled Semi-Relativistic Hartree Equations with Attractive Interactions

We prove the existence and nonexistence of $L^{2}(\mathbb R^3)$-normalized solutions of two coupled semi-relativistic Hartree equations, which arisen from the studies of boson stars and multi-component Bose–Einstein condensates. Under certain condition on the strength of intra-specie and inter-specie interactions, by proving some delicate energy estimates, we give a precise description on the concentration behavior of ground state solutions of the system. Furthermore, an optimal blowing up rate for the ground state solutions of the system is also proved.

Ground state solutions of the non-autonomous Schrödinger–Bopp–Podolsky system

In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + V(x) u + q^2\phi u = f(u)\\ -\Delta \phi + a^2 \Delta ^2 \phi = 4\pi u^2 \end{array}\right. } \hbox { in }{\mathbb {R}}^3. \end{aligned}$$ - Δ u + V ( x ) u + q 2 ϕ u = f ( u ) - Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 in R 3 . By using some original analytic techniques and new estimates of the ground state energy, we prove that this system admits a ground state solution under mild assumptions on V and f. In the final part of this paper, we give a min-max characterization of the ground state energy.

Existence of groundstates for Choquard type equations with Hardy–Littlewood–Sobolev critical exponent

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space $\mathbb{R}^{N}$ R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results.

Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , where $N\geq 3$ N ≥ 3 , $0<\mu <N$ 0 < μ < N , $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ 2 N − μ N ≤ p < 2 N − μ N − 2 , ∗ represents the convolution between two functions. We assume that the potential function $V(x)$ V ( x ) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.