Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian

Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω ∈ C 2 : \Omega \in C^2: ( − Δ ) S p s u ( x ) + u ( x ) = u 2 s ∗ − 1 ( x ) (-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x) . Here ( − Δ ) S p s (-\Delta )_{Sp}^s stands for the s s th power of the conventional Neumann Laplacian in Ω ⋐ R n \Omega \Subset \mathbb {R}^n , n ≥ 3 n \geq 3 , s ∈ ( 0 , 1 ) s \in (0, 1) , 2 s ∗ = 2 n / ( n − 2 s ) 2^*_s = 2n/(n-2s) . For the local case where s = 1 s = 1 , corresponding results were obtained earlier for the Neumann Laplacian and Neumann p p -Laplacian operators.

Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

In this paper we study the Dirac equation with Coulomb potential − i α · ∇ u + a β u − μ | x | u = f ( x , | u | ) u , x ∈ R 3 where a is a positive constant, μ is a positive parameter, α = ( α 1 , α 2 , α 3 ), α i and β are 4 × 4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as μ → 0 + , this result can characterize some relationship of the above problem between μ > 0 and μ = 0.

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 result for critical Klein-Gordon-Maxwell system involving steep potential well

This paper deals with the following system \begin{equation*} \left\{\begin{aligned} &{-\Delta u+ (\lambda A(x)+1)u-(2\omega+\phi) \phi u=\mu f(u)+u^{5}}, & & {\quad x \in \mathbb{R}^{3}}, \\ &{\Delta \phi=(\omega+\phi) u^{2}}, & & {\quad x \in \mathbb{R}^{3}}, \end{aligned}\right. \end{equation*} where $\lambda, \mu>0$ are positive parameters. Under some suitable conditions on $A$ and $f$, we show the boundedness of Cerami sequence for the above system by adopting Poho\v{z}aev identity and then prove the existence of ground state solution for the above system on Nehari manifold by using Br\’{e}zis-Nirenberg technique, which improve the existing result in the literature.

Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

In this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , where $\lambda >0$ λ > 0 is a parameter, $\alpha \in (0,N)$ α ∈ ( 0 , N ) , $N\ge 3$ N ≥ 3 , $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ I α : R N → R is the Riesz potential. As usual, $\alpha /N+1$ α / N + 1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$ λ > λ ∗ for some given number $\lambda _{*}$ λ ∗ in three cases: (i) $2< p<\frac{4}{N}+2$ 2 < p < 4 N + 2 , (ii) $p=\frac{4}{N}+2$ p = 4 N + 2 , and (iii) $\frac{4}{N}+2< p<2^{*}$ 4 N + 2 < p < 2 ∗ . Our result improves the previous related ones in the literature.

A numerical scheme for the ground state of rotating spin-1 Bose–Einstein condensates

We study the existence of nontrivial solution branches of three-coupled Gross–Pitaevskii equations (CGPEs), which are used as the mathematical model for rotating spin-1 Bose–Einstein condensates (BEC). The Lyapunov–Schmidt reduction is exploited to test the branching of nontrivial solution curves from the trivial one in some neighborhoods of bifurcation points. A multilevel continuation method is proposed for computing the ground state solution of rotating spin-1 BEC. By properly choosing the constraint conditions associated with the components of the parameter variable, the proposed algorithm can effectively compute the ground states of spin-1 $$^{87}Rb$$ 87 R b and $$^{23}Na$$ 23 N a under rapid rotation. Extensive numerical results demonstrate the efficiency of the proposed algorithm. In particular, the affect of the magnetization on the CGPEs is investigated.

Existence of multiple nontrivial solutions of the nonlinear Schrödinger-Korteweg-de Vries type system

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.