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In this paper, we consider the following critical fractional magnetic Choquard equation: ε2s(−Δ)A∕εsu+V(x)u=εα−N∫RN∣u(y)∣2s,α∗∣x−y∣αdy∣u∣2s,α∗−2u+εα−N∫RNF(y,∣u(y)∣2)∣x−y∣αdyf(x,∣u∣2)uinRN,\begin{array}{rcl}{\varepsilon }^{2s}{\left(-\Delta )}_{A/\varepsilon }^{s}u+V\left(x)u& =& {\varepsilon }^{\alpha -N}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u(y){| }^{{2}_{s,\alpha }^{\ast }}}{| x-y\hspace{-0.25em}{| }^{\alpha }}{\rm{d}}y\right)| u\hspace{-0.25em}{| }^{{2}_{s,\alpha }^{\ast }-2}u\\ & & +{\varepsilon }^{\alpha -N}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{F(y,| u(y){| }^{2})}{| x-y\hspace{-0.25em}{| }^{\alpha }}{\rm{d}}y\right)\hspace{0.08em}f\left(x,| u\hspace{-0.25em}{| }^{2})u\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array} where ε>0\varepsilon \gt 0, s∈(0,1)s\in \left(0,1), α∈(0,N)\alpha \in \left(0,N), N>max{2μ+4s,2s+α∕2}N\gt {\rm{\max }}\left\{2\mu +4s,2s+\alpha /2\right\}, 2s,α∗=2N−αN−2s{2}_{s,\alpha }^{\ast }=\frac{2N-\alpha }{N-2s} is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, (−Δ)As{\left(-\Delta )}_{A}^{s} stands for the fractional Laplacian with periodic magnetic field AA of C0,μ{C}^{0,\mu }-class with μ∈(0,1]\mu \in (0,1] and VV is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on VV and ff, we establish the existence of at least one ground state solution.

In this paper, we consider the following fourth‐order equation of Kirchhoff type Δ2u−(a+b‖∇u‖22)Δu=λu+|u|p−2uinℝd$$ {\Delta}^2u-\left(a+b{\left\Vert \nabla u\right\Vert}_2^2\right)\Delta u=\lambda u+{\left|u\right|}^{p-2}u\kern0.30em \mathrm{in}\kern6.05pt {\mathrm{\mathbb{R}}}^d $$ under the normalized constraint ‖u‖2=m$$ {\left\Vert u\right\Vert}_2=m $$, where a,b,m>0$$ a,b,m>0 $$ are constants, d≥1$$ d\ge 1 $$ and 2

In this paper, we consider the following fractional Choquard equation: \[ \begin{cases} (-\Delta)^s u+V(x) u+\lambda u=(I_\alpha\ast F(u))f(u)\quad \mathrm{in}\ \mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}} u^{2}\,\mathrm{d} x=\rho^{2},\ \rho>0, \end{cases} \] { (− Δ) s u + V (x) u + λu = (I α ∗ F (u)) f (u) in R N , ∫ R N u 2 d x = ρ 2 , ρ > 0 , where $ s\in (0,1) $ s ∈ (0 , 1) , $ \alpha \in (0,N) $ α ∈ (0 , N) , N>2s and $ V:\mathbb {R}^{N}\rightarrow \mathbb {R} $ V : R N → R is a continuous nonnegative potential and vanishes at infinity. Under some mild assumptions imposed on V and f, we establish the existence of $ L^2 $ L 2 -normalized solution $ (u,\lambda)\in H^{s}(\mathbb {R}^{N})\times \mathbb {R}^{+} $ (u , λ) ∈ H s (R N) × R + . [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger–Poisson system { (− Δ) s u + V (x) u + ϕ u = f (x , u) + λ g (x , u) i n R 3 , (− Δ) α ϕ = u 2 i n R 3 , where λ is a parameter, s , α ∈ (0 , 1) and 2 s + 2 α > 3. Under some suitable assumptions on V, f and g, we obtained the multiplicity of solutions for the above system, by applying variational methods and Ekeland's variational principle. Some recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales , subject to the multi-point boundary condition , where is an increasing homeomorphism and satisfies the relation for , which generalizes the usually p-Laplacian operator. An example applying the result is also presented. The main tool of this paper is a generalization of Leggett-Williams fixed point theorem, and the interesting points are that the nonlinearity f contains the first-order derivative explicitly and the operator is not necessarily odd.