This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i\partial_t u+Δu= θ(ωt)\big(e^{4π|u|^2}-1\big)$, where $θ$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 \in H^1(\mathbb{R}^2)$, the solution $u_ω$ converges, as $|ω|$ tends to infinity to the solution $U$ of the limiting equation $i\partial_t U+ΔU= I(θ)\big(e^{4π|U|^2}-1\big)$ with the same initial data, where $I(θ)$ is the average of $θ$.
We find a solution of a quasilinear elliptic equation with Dirichlet's boundary condition on a smooth bounded domain and involving an unbounded continuous nonlinearity with oscillatory behavior near the origin.
This paper is devoted to the homogenization of Shrödinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and we derive the macroscopic homogenized model. Our approach is the well known two-scale convergence method.
We study the equation $-Δ_g w+w=λα(σ) f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d \geq 3$. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities $f$ which have an oscillating behavior either at zero or at infinity.
We consider a Schrödinger-Poisson system involving a general nonlinearity at critical growth and we prove the existence of positive solutions. The Ambrosetti-Rabinowitz condition is not required. We also study the asymptotics of solutions with respect to a parameter.
This paper mainly discuss the regularity behavior of the hyperbolic magnetic Schroedinger equation with singular coefficients near the origin. We apply the techniques from the microlocal analysis to explore the upper bound of loss of regularity. Furthermore, in order to demonstrate the optimality of the result, a delicate counterexample with periodic coefficients will be constructed to show the lower bound of loss of regularity by the application of harmonic analysis and instability arguments.
The initial value problem is considered for a higher order nonlinear Schrödinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.
We consider the problem of recovering a spatially-localized cubic nonlinearity in a nonlinear Schrödinger equation in dimensions two and three. We prove that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that determines the X-ray transform of the nonlinear coefficient.
We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity $$ i\partial_t u+Δu=\frac{u}{|x|^b}(e^{α|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $α=2π(2-b)$. We establish local and global well-posedness in the subcritical and critical regimes.
We consider the Schrödinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, $δ$, or $δ'$ boundary condition, we show that the non-trivial global solution does not scatter to standing waves. Our proof is based on the argument by Murphy and Nakanishi, who treated the long-range nonlinear Schrödinger equation with a general potential in the Euclidean space, in order to consider general boundary conditions.
We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.
In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity which is of the long range critical order and is not necessarily a polynomial, in one and two space dimensions. As the nonlinearity is the critical order, the possible asymptotic behavior depends on the shape of the nonlinearity. The aim here is to give a sufficient condition on the nonlinearity to construct a modified wave operator. To deal with a non-polynomial nonlinearity, we decompose it into a resonant part and a non-resonant part via the Fourier series expansion. Our…
In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge' function is radially symmetric, w…
In this paper, a nonlinear Schrödinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that both types of wave solutions are orbitally stable under the flow of the equation by minimizing the charge/energy functional.
We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As …
The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term with general nonlinearity: $$ \left\{ \begin{array}{ll} -Δu+V(x)u- ϕ|u|^3u= f(u), & x\in\mathbb{R}^3, -Δϕ= |u|^5, & x\in\mathbb{R}^3,\\ \end{array} \right. $$ Under certain assumptions on non-constant $V(x)$, the existence of a positive least energy solution is obtained by using some new analytical skills and Pohožaev type manifold. In particular, the Ambrosetti-Rabinowitz type condition or monotonicity assumption on the nonlinearity is not necessary.
We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity $λ|u|^αu$, where $0<α<2$, and $λ$ is a complex constant satisfying $\text{Im} λ>\frac{α|\text{Re} λ|}{2\sqrt{ α+1}}$. For arbitrary large initial data, we present the uniform time decay estimates when $4/3<α<2$, and the large time asymptotics of the solution when $\frac{7+\sqrt{145}}{12}<α<2$. The proof is based on the vector fields method and a semiclassical analysis method.
In this paper we consider the following quasilinear Schrödinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - Δu +ϕu = f(u) &\ \mbox{in } Ω, -Δϕ- \varepsilon^{4}Δ_4 ϕ= u^{2} & \ \mbox{in } Ω, u=ϕ=0 & \ \mbox{on } \partialΩ\end{array} \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\varepsilon}, ϕ_{\varepsilon})$ and then we show that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial solution of the associa…
Let $T_ε$ be the lifespan for the solution to the Schrödinger equation on $\mathbb{R}^d$ with a power nonlinearity $λ|u|^{2θ/d}u$ ($λ\in \mathbb{C}$, $0<θ<1$) and the initial data in the form $εφ(x)$. We provide a sharp lower bound estimate for $T_ε$ as $ε\to +0$ which can be written explicitly by $λ$, $d$, $θ$, $φ$ and $ε$. This is an improvement of the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].
In this paper, we study the following fractional Schrödinger equation: \[ \left\{\begin{gathered} {(- Δ)^s}u + mu = f(u){\text{in}}{\mathbb{R}^N}, \hfill u \in {H^s}({\mathbb{R}^N}),{\text{}}u > 0{\text{on}}{\mathbb{R}^N}, \hfill \\ \end{gathered} \right. \] where $m>0$, $N>2s$, ${(- Δ)^s}$, $s \in (0,1)$ is the fractional Laplacian. Using minimax arguments, we obtain a positive ground state solution under general conditions on $f$ which we believe to be almost optimal.
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]}.
We consider the following nonlinear Schrödinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\left(\frac{1}{|x|^{2σ}}\star|u|^2\right)u=0 \] in $\mathbb{R}^N$. We are interested in the existence and behaviour of minimal mass blow-up solutions. Previous studies have shown the existence of minimal mass blow-up solutions with an inverse power potential and investigated the behaviour of the solution. In this paper, we investigate Hartree nonlinearity, which is a nonlinear term similar to the inverse power-type potential in terms of scaling.
This paper is concerned with the upper bound of the lifespan of solutions to nonlinear Schrödinger equations with general homogeneous nonlinearity of the critical order. In [8], Masaki and the first author obtain the upper bound of the lifespan of solutions to our equation via a test function method introduced by [16, 17]. Their nonlinearity contains a non-oscillating term $|u|^{1+2/d}$ which causes difficultly for constructing an even small data global solution. The non-oscillating term corresponds to the $L^1$-scaling critical. In this paper, it turns out that the upper bound can be refined …
In this paper, we study the following fractional Schrödinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-Δ)^su+V(x)u+ϕu=g(u) & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-Δ)^tϕ=u^2,\,\, u>0& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $s,t\in(0,1)$, $\varepsilon>0$ is a small parameter. Under some suitable assumptions on potential function $V(x)$ and critical nonlinearity term $g(u)$, we construct a family of positive solutions $u_{\varepsilon}\in H^s(\mathbb{R}^3)$ which concentrates around the global minima of $V$ as $\varepsilon\right…
We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schrödinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term, in order to weaken the necessary assumptions to obtain a linking structure to the problem, for instance to allow zero being in the spectrum or the nonlinearity being sign-changing. Our main difficulty is to overcome the lack of monotonicity on the nonlinear term, as well, as the lack of compactness since the domain is unbounded. W…
In this paper we consider the following quasilinear Schrödinger-Poisson system $$ \left\{ \begin{array}[c]{ll} - Δu +u+ϕu = λf(x,u)+|u|^{2^{*}-2}u &\ \mbox{in } \mathbb{R}^{3} \\ -Δϕ-\varepsilon^{4} Δ_4 ϕ= u^{2} & \ \mbox{in } \mathbb{R}^{3}, \end{array} \right. $$ depending on the two parameters $λ,\varepsilon>0$. We first prove that, for $λ$ larger then a certain $λ^{*}>0$, there exists a solution for every $\varepsilon>0$. Later, we study the asymptotic behaviour of these solutions whenever $\varepsilon$ tends to zero, and we prove that they converge to the solution of the Schrödinger-Poiss…
In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem \begin{align*} \varepsilon^{2s}(-Δ)^su+V(x)u=f(u) \ \ \ \mbox{in} \ \ \ \mathbb{R}^N, \end{align*} where $N>2s$ and the nonlinearity $f$ has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of $V$ as $\varepsilon\rightarrow 0$. Our result improves the study made in X. He and W. Zou ({\it Calc. Var. Partial Differen…
We study the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus resolving an open problem of thirty years since Bourgain (1993). In view of ill-posedness in negative Sobolev spaces, this result is sharp. We establish a crucial bilinear estimate by separately studying the non-resonant and nearly resonant cases. As a corollary, we obtain a tri-linear version …
We consider the existence and multiplicity of solutions for a class of quasi-linear Schrödinger equations which include the modified nonlinear Schrödinger equations. A new perturbation approach is used to treat the sub-cubic nonlinearity.
