We consider the following coupled fractional Schrödinger system: \begin{equation*} \left\{ \begin{aligned} &(-Δ)^su+λ_1u=μ_1|u|^{2p-2}u+β|v|^p|u|^{p-2}u\\ &(-Δ)^sv+λ_2v=μ_2|v|^{2p-2}v+β|u|^p|v|^{p-2}v\\ \end{aligned} \right.\quad\text{in}~{\mathbb{R}^N}, \end{equation*} with $0<s<1$, $2s<N\le 4s$ and $1+\frac{2s}{N}<p<\frac{N}{N-2s}$, under the following constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2dx=a_1^2\quad\text{and}\quad \int_{\mathbb{R}^N}|v|^2dx=a_2^2. \end{align*} Assuming that the parameters $μ_1,μ_2,a_1, a_2$ are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $β>0$ .
We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.
We are concerned with the nonlinear Schrödinger equation with an $L^2$ mass constraint on both finite and locally finite graphs and prove that the equation has a normalized solution by employing variational methods. We also pay attention to the behaviours of the normalized solution as the mass constraint tends to $0^+$ or $+\infty$ and give clear descriptions of the limit equations. Finally, we provide some numerical experiments on a finite graph to illustrate our theoretical results.
In this paper we study a Schrödinger-Bopp-Podolsky system of partial differential equations in a bounded and smooth domain of $\mathbb R^3$ with a non constant coupling factor. Under a compatibility condition on the boundary data we deduce existence and multiplicity of solutions by means of the Ljusternik-Schnirelmann theory.
We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.
The existence of ground states and (multiple) bound states to semilinear time-independent Maxwell and Schrödinger equations, with or without $L^2$-constraints, is investigated.
In this paper we investigate the existence of positive solutions and ground state solution for a class of fractional Schrödinger-Poisson equations in $\mathbb R^3$ with general nonlinearities.
We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.
We firstly prove Strichartz estimates for the fractional Schrödinger equations on $\mathbb{R}^d$ endowed with a smooth bounded metric $g$. We then prove Strichartz estimates for the fractional Schrödinger and wave equations on compact Riemannian manifolds without boundary $(M,g)$. We finally give applications of Strichartz estimates for the local well-posedness of the pure power-type nonlinear fractional Schrödinger and wave equations posed on $(M,g)$.
In this paper, we study the following energy functional originates from the Schrödinger-Bopp-Podolsky system $$I(u)=\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx+\frac{1}{4}\int_{\mathbb{R}^{3}} ϕ_{u}u^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|u|^{p}dx$$ constrained on $B_ρ=\left\{u\in H^{1}(\mathbb{R}^{3},C):\ \left\|u\right\|_{2}=ρ\right\},$ where $ρ>0.$ As such constrained problem $I(u)$ is bounded from below on $B_ρ$ when $p\in(2,\frac{10}{3}).$ We use minimizing method to get a normalized solution.
In this paper we consider a class of fractional Schrödinger equations with potentials vanishing at infinity. By using a minimization argument and a quantitative deformation Lemma, we prove the existence of a sign-changing solution.
In the present work we are concerned with the existence of normalized solutions to the following Schrödinger-Poisson System $$ \left\{ \begin{array}{ll} -Δu + λu + μ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ } \mathbb{R}^2 , \\ \intR |u(x)|^2 dx = c,\ c> 0 , \end{array} \right. $$ for $μ\in \R $ and a nonlinearity $f$ with exponential critical growth. Here $ λ\in \R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in \cite{Ji} and \cite{[cjj]}.
The purpose of this paper is to establish the existence of solutions with prescribed norm to a class of nonlinear equations involving the mixed fractional Laplacians. This type of equations arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. Our method is novel and our results cover all the previous ones.
We prove the existence of non-trivial solutions for a fractional Schr$\ddot{o}$dinger-Poisson equation in $\mathbb{R}^{3}$. The proof is based on the perturbation method and the mountain pass theorem.
The present article deals with the similarity method to tackle the fractional Schrodinger equation where the derivative is defined in the Riesz sense. Moreover the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives .
In this paper, we consider the Cauchy problem for the fractional Schrödinger equation $i D_t^αu + (-Δ)^{\fracβ{2}} u =0$ with $0<α<1$, $β>0$. We establish the dispersive estimates for the solutions. In particular, we prove that the decay rates are sharp.
We are concerned with the following nonlinear Schrödinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -Δu+λu=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=ρ, \end{cases} \end{aligned} \end{eqnarray*} where $ρ>0$ is given, $λ\in\mathbb{R}$ arises as a Lagrange multiplier and $f$ satisfies an exponential critical growth. Without assuming the Ambrosetti-Rabinowitz condition, we show the existence of normalized ground state solutions for any $ρ>0$. The proof is based on a constrained minimization method and the Trudinger-Moser inequality …
This paper is concerned with the existence of normalized solutions of the nonlinear Schrödinger equation \[ -Δu+V(x)u+λu = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We prove the existence of a solution $(u,λ)\in H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=ρ$ under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.
We consider an optimal rearrangement minimization problem involving the fractional Laplace operator $(-Δ)^s$, $0<s<1$, and Gagliardo-Nirenberg seminorm $|u|_s$. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies $$ -(-Δ)^s U-χ_{\{U\leq 0\}}\min\{-(-Δ)^s U^+;1\}=χ_{\{U>0\}}, $$ which happens to be the fractional analogue of the normalized obstacle problem $Δu=χ_{\{u>0\}}$. A new section analyzing $s \to 1$ has been added.
In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-Δu+λ_{1}u=H_{u}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &-Δv+λ_{2} v=H_{v}(u,v), \quad \quad \hbox{in }\mathbb{R}^{2},\\ &\int_{\mathbb{R}^{2}}|u|^{2}dx=a^{2},\quad \int_{\mathbb{R}^{2}}|v|^{2}dx=b^{2}, \end{aligned} \right. \end{align*} where $a,b>0$ are prescribed, $λ_{1},λ_{2}\in \mathbb{R}$ and the functions $H_{u},H_{v}$ are partial derivatives of a Carathéodory function $H$ with $H_{u},H_{v}$ have exponential g…
In this paper, we are concerned with the ground state solutions of nonlinear fractional Schrödinger equation involving critical growth. Without Ambrosetti-Rabinowitz condition and monotonicity condition on the nonlinearity, we get the existence of ground state solutions of such equation when the potential $V(x)$ is not a constant and not radial.
We investigate the existence of solutions to the fractional nonlinear Schrödinger equation $(-Δ)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly general assumptions on the nonlinearity $f$, we prove the existence of a ground state solution and a multiplicity result in the radially symmetric case.
This paper studies the multiplicity of normalized solutions to the Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -Δu=λu+h(εx)|u|^{q-2}u+η|u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $a, ε, η>0$, $q$ is $L^2$-subcritical, $p$ is $L^2$-supercritical, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $h$ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $ε$ is small enoug…
In this paper we study the existence of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$, $λ\in \mathbb{R}$ and $f$ has an exponential critical growth when $N=2$, and $f(t)=μ|u|^{q-2}u+|u|^{2^*-2}u$ with $q \in (2+\frac{4}{N},2^*)$, $μ>0$ and $2^*=\frac{2N}{N-2}$ when $N \geq 3$. Our main results complement some recent results for $N \geq 3$ and it is totally new for $N=2$.
In this paper, an initial value problem for a nonlinear time-fractional Schrödinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions.
We consider some nonlinear fractional Schrödinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions.
In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation \begin{equation*} \left\{ \begin{array}{l} -Δu + V(x) u = μ_q u + a|u|^q u \quad {\rm in}\quad \mathbb{R}^2,\\ \int_{\mathbb{R}^2}|u|^2\,dx =1,\\ \end{array} \right. \end{equation*} where $μ_q$ is the Lagrange multiplier. We show that for $q>2$ close to $2$, the equation admits two solutions: one is the local minimal solution $u_q$ and another one is the mountain pass solution $v_q$. Furthermore, we study the limiting behavior of $u_q$ and $v_q$ when $q\to 2_+…
This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} Δ^{2}u-λu=μ|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $μ>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ is the $H^2$-critical Sobolev exponent, and $λ$ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are p…
In this paper, we study forward problem and inverse problem for the fractional magnetic Schrodinger equation with nonlinear electric potential. We first investigate the maximum principle for the linearized equation and apply it to show that the problem is well-posed under suitable assumptions on the exterior data. Moreover, we explore uniqueness of recovery of both magnetic and electric potentials.
The semilinear space-time fractional Schrödinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.
We consider a class of parametric Schrödinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.
We deal with a class of fractional magnetic Schrödinger equations in the whole line with exponential critical growth. Under a local condition on the potential, we use penalization methods and Ljusternik-Schnirelmann category theory to investigate the existence, multiplicity and concentration of nontrivial weak solutions.
