We study the following fractional Choquard equation ε 2 s ( − ∆ ) s u + V ( x ) u = ε − µ ( K µ ∗ F ( u ) ) F ′ ( u ) , x ∈ R N , where ε>0 is a small parameter, s∈(0 ,1), N⩾3, µ∈(0 ,N), F ∈ C 1 ( R , R ) , K µ is the Riesz potential. By applying a new variational approach, under some appropriate conditions on V( x), we obtain there exist at least cupl ( V ) + 1 solutions to the above equation when ϵ→0. In addition, we have demonstrated that the concentration behavior of positive solutions occurs around V as ϵ→0, where V is the set where the potential attains its minimum values.

In this paper, we study the following eigenvalue problem for Kirchhoff type equation with Hartree nonlinearity: \begin{equation} -M\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)\Delta u+\mu V(x)u=\left( I_{\alpha }\ast Q\left\vert u\right\vert^{p}\right) Q\vert u\vert ^{p-2}u+\lambda f(x)u \quad \text{in}\quad \mathbb{R}^{N}, \end{equation} where $N\geq3, a, \mu>0$ parameters, $M(t)=at+1$, $V\in C(\mathbb{R}^{N},\mathbb{R}% ^{+}) $, $I_{\alpha }$ is the Riesz potential, $Q(x)\in L^{\infty}(\mathbb{R}^{N})$ with changes sign in $\overline{\Omega}:=\left\{V(x)=0\right\}$, and $0

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation involving the fractional p-Laplace operator ( − Δ)ps. More precisely, we consider $$ M\left(^{2N}}}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s} u=\lambda f(x)|u|^{q-2}u+K(x)|u|^{p_{s}^{*}-2}u,\quad &{\rm in}\ \Omega,\\ u=0, \quad\quad &{\rm in}\ ^{N}\setminus \Omega, \\ $$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary ∂Ω, M(t)=a + btm − 1 with m > 1, a > 0, b > 0, dimension N > sp, $ p_{s}^{*}={N-ps}$ is the fractional critical Sobolev exponent, and the parameters λ > 0, 0 < s < 1 < q < p < ∞. Applying Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity of nonnegative solutions.

In this paper, we study the following nonlinear Schrödinger-Bopp-Podolsky system: −∆u + u + l(x)φu = a(x)|u|p−2u + µb(x)|u|q−2u + |u|5, in R3, −∆φ + a2∆2φ = l(x)u2, in R3, where p,q ∈ (4,6), µ > 0, l(x), a(x) and b(x) are nonnegative continuous functions. Under some certain assumptions, we prove the above system have ground state and multiple solutions by using variational.