On a Fractional Schrödinger–Poisson System with Doubly Critical Growth and a Steep Potential Well
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Cited by 3 publications
References 45 publications
Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: ε 2 s ( − Δ ) s u + V ( x ) u + ϕ u = H ( u − β ) f ( u ) , in R 3 , ε 2 s ( − Δ ) s ϕ = u 2 , in R 3 , u > 0 , in R 3 , $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ u{ >}0,\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where ɛ > 0 is a small parameter, s ∈ ( 3 4 , 1 ) $s\in \left(\frac{3}{4},1\right)$ , β > 0, H is the Heaviside function, (−Δ) s u is the fractional Laplacian operator, V : R 3 → R $V :{\mathbb{R}}^{3}\to \mathbb{R}$ is a continuous potential and f : R → R $f :\mathbb{R}\to \mathbb{R}$ is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate.
Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity
In this paper, we study the following fractional Schrödinger–Poisson system with discontinuous nonlinearity: ε 2 s ( − Δ ) s u + V ( x ) u + ϕ u = H ( u − β ) f ( u ) , in R 3 , ε 2 s ( − Δ ) s ϕ = u 2 , in R 3 , u > 0 , in R 3 , $$\begin{cases}^{2s}{\left(-{\Delta}\right)}^{s}u+V\left(x\right)u+\phi u=H\left(u-\beta \right)f\left(u\right),\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ {\varepsilon }^{2s}{\left(-{\Delta}\right)}^{s}\phi ={u}^{2},\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ u{ >}0,\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where ɛ > 0 is a small parameter, s ∈ ( 3 4 , 1 ) $s\in \left(\frac{3}{4},1\right)$ , β > 0, H is the Heaviside function, (−Δ) s u is the fractional Laplacian operator, V : R 3 → R $V :{\mathbb{R}}^{3}\to \mathbb{R}$ is a continuous potential and f : R → R $f :\mathbb{R}\to \mathbb{R}$ is superlinear continuous nonlinearity with subcritical growth at infinity. By using nonsmooth analysis, we investigate the existence and concentration of solutions for the above problem. Moreover, we obtain some properties of these solutions, such as convergence and decay estimate.
Concentrating Solutions for Fractional Schrödinger–Poisson Systems with Critical Growth
In this paper, we consider a class of fractional Schrödinger–Poisson systems (−Δ)su+λV(x)u+ϕu=f(u)+|u|2s*−2u and (−Δ)tϕ=u2 in R3, where s,t∈(0,1) with 2s+2t>3, λ>0 denotes a parameter, V:R3→R admits a potential well Ω≜intV−1(0) and 2s*≜63−2s is the fractional Sobolev critical exponent. Given some reasonable assumptions as to the potential V and the nonlinearity f, with the help of a constrained manifold argument, we conclude the existence of positive ground state solutions for some sufficiently large λ. Upon relaxing the restrictions on V and f, we utilize the minimax technique to show that the system has a positive mountain-pass type by introducing some analytic tricks. Moreover, we investigate the asymptotical behavior of the obtained solutions when λ→+∞.
Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well
<abstract><p>In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.</p></abstract>
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