Yanjun Liu, Shijie Qi
In this paper, we consider Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4,using singular Adams inequality and variational techniques, we get the existence of ground state solutions. Moreover, we also get the same results without the Ambrosetti-Rabinowitz (AR) condition.
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In this article, we establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz condition.
In this paper we consider a generalized Kirchhoff? equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools.
Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
本文针对有界区域上含分数阶$p$-Laplace算子的临界Schrödinger-Kirchhoff型系统，结合Nehari流形上能量泛函性质与纤维映射分析，建立了该系统解的多解性结果。
Existence of solutions to four-dimensional Kirchhoff problems with critical-concave nonlinearities
研究\(\mathbb{R}^4\)中有竞争临界和凹项的基尔霍夫问题，用新逼近技术和流形分析得出存在性结果，补充临界凸情形，给出解存在时参数范围的明确界限。
Fractional p-Kirchhoff equation with Sobolev and Choquard singular nonlinearities
本文研究全空间上含双重非线性项与临界奇异权重的分数阶p-Kirchhoff方程，利用Cerami条件克服慢增长非线性项带来的紧性证明困难，得到非平凡弱解的存在性结果。
In present paper, we study the normalized solutions $(λ_c, u_c)\in \R\times H^1(\R^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\R^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3 $$ satisfying the normalization constraint $ \displaystyle\int_{\R^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c…
In this paper, we consider the multiplicity of solutions for a class of Kirchhoff type problems with sub-linear and critical terms on an unbounded domain. With the aid of Ekeland's variational principle and the concentration compactness principle we prove that the Kirchhoff problem has at least two solutions.
In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable co…
In this article we study modified Kirchhoff-Schrödinger equations involving critical Stein-Weiss type nonlinearity and $p$-Laplacian
Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities
本文研究了大型光滑星形区域及全空间上具有混合非线性项的 Kirchhoff 方程的给定质量解，分析了其存在性、多重性与渐近行为，并指出势函数的存在导致标准 Pohozaev 恒等式方法失效。
In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_Ω|\nabla u|^n)Δ_n u = f(x,u) \; \text{in}\; Ω,\quad u =0\quad\text{on} \quad \partial Ω, \end{array} \right.$$ where $Ω\subset \mathbf{R}^n$ is a bounded domain with smooth boundary, $n\geq 2$ and $f$ behaves like $e^{|u|^{\frac{n}{n-1}}}$ as $|u|\to\infty$. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when $f(x,t)$ is concave near $t=0$ and convex a…
In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u=λu+μ|u|^{q-2}u+|u|^{p-2}u \quad \hbox{in $\mathbb{R}^3$,} \end{equation*} with a prescribed mass \begin{equation*} \int_{\mathbb{R}^{3}}|u|^{2}dx=c^{2}, \end{equation*} where $s\in(0, 1)$, $a, b, c>0$, $2<q<p<2_{s}^{\ast}=\frac{6}{3-2s}$, $μ>0$ and $λ\in\mathbb{R}$ as a Lagrange multiplier. Under different assumptions on $q<p$, $c>0$ and $μ>0$, we prove some existence results about…
In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-Δ)^{s}_p u &=λf(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} Ω,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus Ω, \end{array} \end{equation*} where $(-Δ)^{s}_p$ is the fractional $p$-Laplace operator, $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(Ω)$ and $g\in L^\infty(Ω)$ are sig…
Nontrivial solutions for a $(p,q)$-Kirchhoff type system with concave-convex nonlinearities on locally finite graphs
本文在局部有限图上研究带扰动项的$(p,q)$-Kirchhoff型系统，利用山路定理、Ekeland变分原理和Clark定理证明了完全非平凡解与半平凡解的存在性及多重性，并给出解不存在的条件。
In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: -\left(a+b\ds\int_{\R^3}|D u|^2\right)Δu+u=f(x,u)+u^{5}, u\in H^1(\R^3), u>0, $x\in \R^3$ where a,b>0 are constants. Under certain assumptions on the sign-changing function $f(x,u)$, we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [17] concerning the existence of positive solutions to the nonlinear Kirchhoff problem \left(\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D u|^2\right)Δu+…
In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.
\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity $$ \quad \left\{ \begin{array}{lr} \quad -M\left(\displaystyle\int_Ω|\nabla^m u|^{\frac{n}{m}}dx\right)Δ_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^α} \; \text{in}\; \Om{,} \quad \quad u = \nabla u=\cdot\cdot\cdot= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Om{,} \end{array} \right. $$ where $\Om\subset \mb R^n$ is a bounded domain with smooth boundary, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$…
Three weak solutions for a $(p, q)$-Schrödinger-Kirchhoff type equation
该研究针对满足del Pino-Felmer型条件的含连续正势的(p, q)-Schrödinger-Kirchhoff方程，运用Recceri经典变分方法证明了其三个弱解的存在性。
In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schrödinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.
We consider the cubic nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure of scattering for the Schrödinger equation with power-type long-range nonlinearities. These results are extension of the results for NLS on the one dimensional Euclidean space.
In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff-Schrödinger-Poisson system involving the weak singularity, i.e. when $0<γ<1$. Further, we obtain the existence of a solution with the strong singularity, i.e. when $γ>1$. We employ variational techniques to prove the existence and multiplicity results. Moreover, a $L^{\infty}$ estimate is obtained by using the Moser iteration method.
In this paper we establish the existence of mountain pass and negative energy weak solutions for a Kirchhoff-Schrödinger type problem in $\mathbb R^4$ involving a critical nonlinearity and a suitable small perturbation. The arisen competition between the terms due to the nonlocal coefficient and critical nonlinearity turns out to be rather interesting. The main tools used in the present work are variational methods and the Lions' Concentration Compactness Principle.
In this note, we deal with a problem of the type $$\cases {-h\left ( \int_Ω|\nabla u(x)|^2dx\right ) Δu=f(u) & in $Ω$\cr & \cr u_{|\partialΩ}=0\ .\cr}$$ As an application of a new general multiplicity result, we establish the existence of at least three solutions, two of which are global minima of the related energy functional. The only condition assumed on $f$ is that it be subcritical and superlinear: no condition on the behaviour of $f$ at $0$ is requested.
We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.
Normalized solution for Kirchhoff equation with upper critical exponent and mixed Choquard type nonlinearities
本文研究了具有混合Choquard类型非线性和上临界指数的Kirchhoff方程的归一化解存在性，通过在L^2约束下进行精细能量估计恢复紧性，并克服双重非局部项困难，推广了Wang等人关于低指数区间的结果。
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-Δ)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-Δ)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential …
In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems in unbounded domains, which involves a general variable exponent elliptic operator. Under some suitable conditions on the nonlinearities, we establish existence of at least three weak solutions for the problem. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano.
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
Fractional p-Laplacian Kirchhoff-type problem involving a singular term via Nehari manifold
本文利用Nehari流形和Ekeland变分原理，证明了含变号非线性与奇异项的分数阶p-Laplacian Kirchhoff型问题在次临界和临界情形下至少存在两个正解。
On the Kirchhoff equation with prescribed mass and general nonlinearities
本文用全局分支方法研究带质量和一般非线性的Kirchhoff方程正归一化解的存在性、不存在性与多重性，首次处理高维（N≥5）因非局部项导致的缺乏山路几何情形。
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.
In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)Δu+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in \R^3, where $a,$ $b>0$ are constants, $2<p<5$ and $V:\R^3\rightarrow\R$. Under certain assumptions on $V$, we prove that \eqref{0.1} has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results can be viewed as a partial extension of a recent result of He and Zou in [16] concerning the existence of positive solutions to the nonlinear Kirchhof…
Using a new infinite-dimensional linking theorem, we obtained nontrivial solutions for strongly indefinite periodic Schrödinger equations with sign-changing nonlinearities.
The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $θ\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-Δ)_A^s$ is the fractional magnetic o…
This paper is devoted to a comprehensive study of the nonlinear Schrödinger equations with combined nonlinearities of the power-type and Hartree-type in any dimension n\ge3. With some structural conditions, a nearly whole picture of the interactions of these nonlinearities in the energy space is given. The method is based on the Morawetz estimates and perturbation principles.
In this article we study the existence of solutions to the system \begin{equation*}\left\{ \begin{array}{ll} -\left(a+b\int_Ω|\nabla u|^{2}\right)Δu +ϕu= f(x, u) &\text{in }Ω\hbox{} -Δϕ= u^{2} &\text{in }Ω\hbox{} u=ϕ=0&\text{on }\partialΩ, \hbox{} \end{array} \right. \end{equation*} where $Ω$ is a bounded smooth domain of $\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\geq0$, and $f:\overlineΩ\times \mathbb{R}\to\mathbb{R}$ is a continuous function which is $3$-superlinear. By using some variants of the mountain pass theorem established in this paper, we show the existence of three solutions: one p…
We consider the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-Δ)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $ u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-Δ)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are t…
We prove the existence of non-trivial solutions to a system of coupled, nonlinear, Schroedinger equations with general nonlinearity.
Recent advances on Schrödinger equations with dissipative nonlinearities
本文综述了带耗散结构的非线性薛定谔方程的最新进展，基于作者近期工作总结了该领域的研究成果。
Infinitely many solutions for $(p,q)$-Schrödinger-Poisson system with concave and convex nonlinearities
本文研究由p-Laplacian Schrödinger方程与q-Laplacian Poisson方程耦合、含凹凸非线性项和不定权函数的拟线性系统，证明了其存在无穷多个解。
In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{…
The semi-classical regime of standing wave solutions of a Schrödinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.
In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region princip…
We study the equation $-Δ_g w+w=λα(σ) f(w)$ on a $d$-dimensional homogeneous Cartan-Hadamard Manifold $\mathcal{M}$ with $d \geq 3$. Without using the theory of topological indices, we prove the existence of infinitely many solutions for a class of nonlinearities $f$ which have an oscillating behavior either at zero or at infinity.
In this work we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms on the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.
We obtain the existence of ground state solution for the nonlocal problem $$ m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-Δu + b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a Kirchhoff-type function, $b$ may be negative and noncoercive, $A$ is locally bounded and the function $f$ has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case $m\equiv 1$. In the proofs we apply Variational Methods beside a new Trudinger-Moser type inequality.
Generalized noncooperative Schrödinger-Kirchhoff-type systems in $\mathbb{R}^N$
本文研究含一般变指数临界增长椭圆算子的非合作 Schrödinger-Kirchhoff 型系统，结合极限指标理论、加权变指数 Sobolev 集中紧性原理与对称临界性原理，证明其存在无穷多解。
We investigate the stability of ground states to a nonlinear focusing Schrödinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability estimate of ground states in the spirit of one due to Weinstein [{\it SIAM J. Math. Anal.}, 16(1985),472-491].
Existence and Multiplicity of Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains
本文利用变分方法研究有界区域中具一般非线性项的薛定谔方程的归一化解，得到两个正归一化解，证明星形有界区域解的多重性及球域中非径向归一化解的存在性。
Consider the initial value problem for systems of cubic derivative nonlinear Schrödinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under which the small data solution gains an additional logarithmic decay as $t \to +\infty$ compared with the corresponding free evolution.
In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u = (|x|^{-α}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-Δ)_{A}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s)$, $α< \min\{N, 4s\}$, $M : \mathbb{R}^{+}_{0}\rightarrow …
We consider the stationary one dimensional Schrödinger-Poisson system on a bounded interval with a background potential describing a quantum well. Using a partition function which forces the particles to remain in the quantum well, the limit $h\rightarrow0$ in the nonlinear system leads to a uniquely solved nonlinear problem with concentrated particle density. It allows to conclude about the convergence of the solution.
On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields
该研究针对带临界非线性项与电磁场的三维退化分数阶Schrödinger-Kirchhoff-Poisson方程，利用集中紧性原理、变分法及精细分析工具，证明了参数趋于零且Kirchhoff项在零点退化时解的存在性与多重性。
Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach
该研究采用全局分岔方法，探究有界区域中具一般非线性项的薛定谔方程指定质量正解的存在性等性质，推广了已有自治齐次非线性情形的结果，并得到相关轨道稳定性结论。
We propose existence and multiplicity results for the system of Schrödinger equations with sign-changing nonlinearities in bounded domains or in the whole space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the space $\mathbb{R}^N$ we additionally need to assume the $\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.
In this work, we show the existence of ground state solutions for an $l$-component system of non-linear Schrödinger equations with quadratic-type growth interactions in the energy-critical case. They are obtained analyzing a critical Sobolev-type inequality and using the concentration-compactness method. As an application, we prove the existence of blow-up solutions of the system without the mass-resonance condition in dimension six (and five), when the initial data is radial.
In this paper, we prove the decay and scattering in the energy space for nonlinear Schrödinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + Δu - V(x)u + λ|u|^{p - 1}u = 0$. We will prove decay estimate and scattering of the solution in the small data case when $1+\frac{2}{d}<p\le1+\frac{4}{d-2}$, $d\ge3$. The index $1+\frac{2}{d}$ is sharp for scattering concerning the result of W. Strauss [21].
We consider the following Schrödinger equation $$ - \hslash ^2 Δu + V(x)u = Γ(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $Γ$ attains global maximum at the same point or $V$ attains global minimum and $Γ$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash>0$.
We study the Schrödinger equation: \begin{eqnarray} - Δu+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of the operator $-Δ+V$. We prove that under some new assumptions for $f$, this equation has a nontrivial solution. Our assumptions for the nonlinearity $f$ are very weak and greatly different from the known assumptions in the literature.
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