In this manuscript, we consider the logarithmic Schrödinger equation
\begin{eqnarray*} -\varepsilon^2Δu+V(x)u=u\log u^{2},\,\,\,u>0, & \text{in}\,\,\,\mathbb{R}^{N}, \end{eqnarray*} where $N\geq3$, $\varepsilon>0$ is a small parameter. Under some assumptions on $V(x)$, we show the existence of positive multi-peak solutions by Lyapunov-Schmidt reduction. It seems to be the first time to study singularly perturbed logarithmic Schrödinger problem by reduction. And here using a new norm is the crucial technique to overcome the difficulty caused by the logarithmic nonlinearity. At the same time, we consider the local uniqueness of the multi-peak solutions by using a type of local Pohozaev identities.
In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schrödinger equation in presence of a smooth external potential.
We study a class of logarithmic Schrodinger equations with periodic potential which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions.
By means of non-smooth critical point theory we obtain existence of infinitely many weak solutions of the fractional Schrödinger equation with logarithmic nonlinearity. We also investigate the Hölder regularity of the weak solutions.
In this paper we present a proof of the orbital stability of ground state for logarithmic Schrödinger equation in any dimension and under nonradial perturbations.
We consider the one-dimensional logarithmic Schrödinger equation with a delta potential. Global well-posedness is verified for the Cauchy problem in H1(R) and in an appropriate Orlicz space. In the attractive case, we prove orbital stability of the ground states via variational approach.
In this paper, we study an inverse coefficients problem for two coupled Schrödinger equations with an observation of one component of the solution. The observation is done in a nonempty open subset of the domain where the equations hold. A logarithmic type stability result is obtained. The main method is based on the Carleman estimate for coupled Schrödinger equations and coupled heatn equations, and the Fourier-Bros-Iagolnitzer transform.
In dimension two, we investigate a free energy and the ground state energy of the Schrödinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrödinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic gr…
We are interested in a WKB analysis of the Logarithmic Non-Linear Schrödinger Equation with "Riemann-like" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross-Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to $0$ at infinity.
In this paper we consider a class of logarithmic Schrödinger equations with a potential which may change sign. When the potential is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem, and in the case of bounded potential we obtain a ground state solution, i.e. a nontrivial solution with least possible energy. The functional corresponding to the problem is the sum of a smooth and a convex lower semicontinuous term.
In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.
In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $\cap$ W 2,$\infty$ and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $\in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.
We study spatially periodic logarithmic Schrödinger equations: \begin{equation}\tag{LS} -Δu + V(x)u=Q(x)u\log u^2, \quad u>0\quad \text{in}\ \mathbb{R}^N, \end{equation} where $N\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\mathbb{Z}^N$-action.
In this article we use the variational method developed by Szulkin \cite{szulkin} to prove the existence of a positive solution for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0, N \geq 1$ and $V$ is a saddle-like potential.
In this paper, we consider the following fractional logarithmic Schrödinger equation \begin{equation*} \varepsilon^{2s}(-Δ)^s u + V(x)u=u\log |u|^2\ \ \text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.
This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0$, $N \geq 1$ and $V$ is a continuous function with a global minimum. Using variational method, we prove that for small enough $ε>0$, the "shape" of the graph of the function $V$ affects the number of nontrivial solutions.
In this article we are concerned with the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0, N \geq 1$ and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential. Under a local assumption on the potential $V$, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.
We study the existence and concentration behavior of the bound states for the following logarithmic Schrödinger equation \begin{equation*} \begin{cases} -\varepsilon^2Δv+V(x)v=v\log v^2 \ \ &\text {in}\ \ \mathbb R^N,\\ v(x)\to 0 \ \ &\text {as}\ \ |x|\to\infty, \end{cases} \end{equation*} where $N\geq 1$, $\varepsilon>0$ is a small parameter, and $V$ may be unbounded below at infinity with a speed of at most quadratic strength. We show that around various types of local topological critical points of the potential function, positive bound state solutions exist and concentrate as $\varepsilon\…
In this paper, we study the following logarithmic Schrödinger equation \[ -Δu+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $Δ$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to \mathbb{R}$ is bounded from below and may change sign. We first establish two Sobolev compact embedding theorems in the case when different assumptions are imposed on $a(x)$. It leads to two kinds of associated energy functionals, one of which is not well-defined under the logarithmic nonlinearity, while the other is $C^1$. The existence of ground state solutions are then obta…
We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-li…
A class of self-similar solutions to the derivative nonlinear Schrödinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions.
In this paper we consider the nonlinear fractional logarithmic Schrödinger equation. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we prove that the set of minimizers is a stable set for the initial value problem, that is, a solution whose initial data is near the set will remain near it for all time.
In this paper, we study the existence, uniqueness, nondegeneracy and some qualitative properties of positive solutions for the logarithmic Schrödinger equations: \[ -Δu+ V(|x|) u=u\log u^2, u\in H^1(\mathbb R^N). \] Here $N\geq 2$ and $V\in C^2((0,+\infty))$ is allowed to be singular at $0$ and repulsive at infinity (i.e., $V(r)\to-\infty$ as ${r\to\infty}$). Under some general assumptions, we show the existence, uniqueness and nondegeneracy of this equation in the radial setting.Specifically, these results apply to singular potentials such as $V(r)=α_{1}\log r+α_2 r^{α_3}+α_4$ with $α_1>1-N$,…
This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -Δu+ λV(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in H^1(\mathbb{R}^{N}), \\ \end{array} \right. $$ where $N \geq 1$, $λ>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ has a potential well $Ω: =\text{int}\, V^{-1}(0)$ which possesses $k$ disjoint bounded components $Ω=\bigcup_{j=1}^{k}Ω_{j}$. Using the variational methods, we prove that if the parameter $λ>0$ is large enough, then …
We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson-a time-independent Gaussian function-is an orbitally stable solution. In the general case in dimension $d = 1$, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the…
We consider the logarithmic Schr{ö}dinger equation (logNLS) in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson-a time-independent Gaussian function-is an orbitally stable solution. In this paper, we construct multi-solitons (or multi-Gaussons) for logNLS, with estimates in $H^1 \cap \mathcal{F} (H^1)$. We also construct solutions to logNLS behaving (in $L^2$) like a sum of $N$ Gaussian solutions with different speeds (which we call multi-gaussian). In both cases, the convergence (as $t \rightarrow \infty$) is faster than exponential. W…
Ground states and sharp logarithmic Sobolev inequalities for logarithmic Schrödinger equations with partial confinement
Jilin University（吉林大学）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Nicolaus Copernicus University（尼古拉·哥白尼大学）；University of Warsaw（华沙大学）
In this paper we study the one-dimensional logarithmic Schrödinger equation perturbed by an attractive $δ^{\prime}$-interaction \[ i\partial_{t}u+\partial^{2}_{x}u+ γδ^{\prime}(x)u+u\, \mbox{Log}\left|u\right|^{2}=0, \quad (x,t)\in\mathbb{R}\times\mathbb{R}, \] where $γ>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $δ^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0<γ\leq 2$, then there is a single ground state and it is an odd function;…
In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schrödinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schrödinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal Hölder regularity estimates and uniform estimates in energy space $\mathbb H^1(\mathcal O)$ and weighted Sobolev space $L^2_α(\mathcal O)$ of the solutions for both SlogS equation and RSlogS equati…
We prove existence results of complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory.
We study the logarithmic Schrödinger equation with finite range potential on $\mathbb{R}^{\mathbb{Z}^d}$. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.
We consider the Cauchy problem of the two-dimensional Schrödinger-Poisson system in the energy class. Though the Newtonian potential diverges at the spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to apply the perturbation method. Our argument can be adapted to the one-dimensional problem.
In this note, we study the integrodifferential operator $(I-Δ)^{\log}$ corresponding to the logarithmic symbol $\log(1+|ξ|^2)$, which is a singular integral operator given by $$ (I-Δ)^{\log} u(x)=d_{N}\int_{\mathbb{R}^N}\frac{u(x)-u(x+y)}{|y|^{N}}ω(|y|)\, dy, $$ where $d_N=π^{-\frac{N}{2}}$, $ω(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}K_{\frac{N}{2}}(r)$ and $K_ν$ is the modified Bessel function of second kind with index $ν$. This operator is the Lévy generator of the variance gamma process and arises as derivative $\partial_s\Big|_{s=0}(I-Δ)^s$ of fractional relativistic Schrödinger operators at $s…
We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schrödinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schrödinger equations on…
In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical nonlinearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik-Schnirelmann category, we deeply investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets rel…
In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schrödinger-Newton system. Precisely, we give the precise description of the chemical potential $μ$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local …
