Showing papers by "Binlin Zhang published in 2020"

Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

Abstract: Abstract This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: - ( a + b ∫ ℝ 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = ( I α * F ( u ) ) f ( u ) , x ∈ ℝ 3 , u ∈ H 1 ( ℝ 3 ) , -\\bigg{(}a+b\\int_{\\mathbb{R}^{3}}\\lvert\ abla u|^{2}\\,\\mathrm{d}x\\bigg{)}% \\Delta u+V(x)u=(I_{\\alpha}*F(u))f(u),\\quad x\\in{\\mathbb{R}}^{3},\\,u\\in H^{1}(% \\mathbb{R}^{3}), where a , b > 0 {a,b>0} , I α : ℝ 3 → ℝ {I_{\\alpha}\\colon\\mathbb{R}^{3}\\rightarrow\\mathbb{R}} , with α ∈ ( 0 , 3 ) {\\alpha\\in(0,3)} , is the Riesz potential, V ∈ 𝒞 ⁢ ( ℝ 3 , [ 0 , ∞ ) ) {V\\in\\mathcal{C}(\\mathbb{R}^{3},[0,\\infty))} , f ∈ 𝒞 ⁢ ( ℝ , ℝ ) {f\\in\\mathcal{C}(\\mathbb{R},\\mathbb{R})} and F ⁢ ( t ) = ∫ 0 t f ⁢ ( s ) ⁢ d s {F(t)\\kern-1.0pt=\\kern-1.0pt\\int_{0}^{t}f(s)\\,\\mathrm{d}s} . By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f. Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case where f ⁢ ( u ) = | u | q - 2 ⁢ u {f(u)=|u|^{q-2}u} , with q ∈ ( 1 + α / 3 , 2 ] {q\\in(1+\\alpha/3,2]} .

Fractional magnetic Schr\"{o}dinger-Kirchhoff problems with convolution and critical nonlinearities.

Abstract: In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-\Delta)_{A}^su + V(x)u = (|x|^{-\alpha}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-\Delta)_{A}^s$ is the fractional magnetic operator with $0 0$ is a positive parameter. The electric potential $V\in C(\mathbb{R}^N, \mathbb{R}^+_0)$ satisfies $V(x) = 0$ in some region of $\mathbb{R}^N$, which means that this is the critical frequency case. We first prove the $(PS)_c$ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term $M$ can vanish at zero.

Homoclinic solutions for Hamiltonian systems with variable-order fractional derivatives

Abstract: In this paper, we are concerned with the multiplicity and concentration of solutions for a Hamiltonian system driven by the fractional Laplace operator with variable order derivative. More precisel...

Variational analysis for nonlocal Yamabe-type systems

Abstract: The paper is concerned with existence, multiplicity and asymptotic behavior of (weak) solutions for nonlocal systems involving critical nonlinearities. More precisely, we consider \begin{document}$\left\{ \begin{array}{*{35}{l}} \begin{align} & M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ & =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ & {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} & \ & \ & {} \\\end{array} \right.$\end{document} where \begin{document}$ (-\Delta )^s $\end{document} is the fractional Lapalcian, \begin{document}$ [u]_{s} $\end{document} is the Gagliardo seminorm of \begin{document}$ u $\end{document} , \begin{document}$ M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0 $\end{document} is a continuous function satisfying certain assumptions, \begin{document}$ V(x) = {|x|^{-2s}} $\end{document} is the Hardy potential function, \begin{document}$ 2_{s, t}^* = {(3+2t)}/{(3-2s)} $\end{document} , \begin{document}$ s, t\in (0, 1) $\end{document} , \begin{document}$ \lambda, \mu $\end{document} are two positive parameters, \begin{document}$ 1 and \begin{document}$ h\in L^{{2_s^*}/{(2_s^*-p)}}(\mathbb{R}^3) $\end{document} . By using topological methods and the Krasnoleskii's genus theory, we obtain the existence, multiplicity and asymptotic behaviour of solutions for above problem under suitable positive parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \mu $\end{document} . Moreover, we also consider the existence of nonnegative radial solutions and non-radial sign-changing solutions. The main novelties are that our results involve the possibly degenerate Kirchhoff function and the upper critical exponent in the sense of Hardy–Littlehood–Sobolev inequality. We emphasize that some of the results contained in the paper are also valid for nonlocal Schrodinger–Maxwell systems on Cartan–Hadamard manifolds.

A multiplicity result for a class of fractional p-Laplacian equations with perturbations in ℝN

Abstract: This paper deals with a class of nonlinear elliptic equations with perturbations in the whole space involving the fractional p-Laplacian. As a particular case, we investigate the following Schrodin...