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.

Existence of Multispike Positive Solutions for a Nonlocal Problem in ℝ3

In this paper, we study the following nonlinear Choquard equation −ϵ2Δu+Kxu=1/8πϵ2∫ℝ3u2y/x−ydyu,x∈ℝ3, where ϵ>0 and Kx is a positive bounded continuous potential on ℝ3. By applying the reduction method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of Kx if ϵ>0 is sufficiently small under some suitable conditions on Kx.

On Nodal Solutions of the Nonlinear Choquard Equation

This paper deals with the general Choquard equation-\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N},where {V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and {I_{\alpha}} denotes the Riesz potential of order {\alpha\in(0,N)}. In view of the convolution term, the nonlocal property makes the variational functional completely different from the one for local pure power-type nonlinearity. By combining the Brouwer degree and developing some new techniques, a family of radial solutions with a prescribed number of zeros is constructed for {p\in[2,\frac{N+\alpha}{N-2})}, while the degeneracy happens for {p\in(\frac{N+\alpha}{N},2)}. This result complements and improves the ones in the literature in the aspect of the range of p.

Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).

Existence of solutions for critical Choquard equations via the concentration-compactness method

In this paper, we consider the nonlinear Choquard equation $$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$ where 0 < μ < N, N ⩾ 3, g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and $G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potential V(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.