Benjamin Dodson
In this paper we prove global well-posedness for the defocusing, energy-subcritical, nonlinear wave equation on $\mathbb{R}^{1 + 3}$ with initial data in a critical Besov space. No radial symmetry assumption is needed.
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This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.
We consider a fractional Schrödinger-Poisson system with a general nonlinearity in subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.
In this paper, we prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by Cuccagna, Georgiev, Visciglia to overcome the weak decay with respect to $t$ of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by Staffilani to c…
We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.
Let us consider a semilinear boundary value problem $ - Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $ Ω\subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions guarantying that semi-stable weak positive solutions to subcritical semilinear elliptic equations are smooth in any dimension, and as a consequence, classical solutions. By a subcritical nonlinearity we mean $f(x,s)/s^\frac{N+2}{N-2} \to 0$ as $s\to\infty$, including non-power nonlinearities, and enlarging the class of subcritical nonlinearities, which is usually reserved for power …
NLS with mass-subcritical combined nonlinearities: small mass $L^2$-scattering
本文研究了具有聚焦主导项与低阶散焦扰动项的双重非线性NLS方程，利用伪共形变换与变分论证证明了仅需初始数据质量足够小即可实现质量次临界区域中的散射。
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.
Liouville theorem for subcritical nonlinear heat equation
研究亚临界非线性热方程的非负古老解，利用Li-Yau型估计和Melre-Zaag结果证明了该方程的利普希茨定理。
Existence of multi-solitons for logarithmic Schr{ö}dinger equation with a perturbation of a smooth $L^2$-subcritical function nonlinearity
本文针对含聚焦对数项和光滑 $L^2$-次临界函数非线性的 Schrödinger 方程，通过迭代构造与局部能量估计，证明了多孤子解的存在性，其渐近表现为多个不同速度孤子的叠加。
In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional $H^s$ scaling subcritical case with $1 \leq s \leq 2$, the local well-posedness follows from a Strichartz estimate. In higher dimensional $H^1$ scaling subcritical case, the local well-posedness for radial solutions follows from a weighted Strichartz estimate. Moreover, in three dimensional $H^1$ scaling critical case, the local well-posedness for radial solutions follows from a uniform bound of solutions which may be derived by the correspond…
In this paper, we introduce two minimization problems on non-scattering solutions to nonlinear Schrödinger equation. One gives us a sharp scattering criterion, the other is concerned with minimal size of blowup profiles. We first reformulate several previous results in terms of these two minimizations. Then, the main result of the paper is existence of minimizers to the both minimization problems for mass-subcritical nonlinear Schrödinger equations. To consider the latter minimization, we consider the equation in a Fourier transform of generalized Morrey space. It turns out that the minimizer …
In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical…
We consider a class of nonlinear Schrödinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time depe…
We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in weighted $L^2$ space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all non-scattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing-wave solution.
In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criteria is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of zero solution and a linear solution. Taking t…
We investigate a class of nonlinear nonautonomous scalar field equations with fractional diffusion, critical power nonlinearity and a subcritical term. The involved potentials are allowed for vanishing behavior at infinity. The problem is set on the whole space and compactness issues have to be tackled.
In this article we present some results on the existence of positive and ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. We introduce a general nonlinearity with subcritical and supercritical growth which does not require the usual Ambrosetti-Rabinowitz condition. The proof is based on variational methods and perturbation arguments.
We study the time-asymptotic behavior of solutions of the Schrödinger equation with nonlinear dissipation \begin{equation*} \partial _t u = i Δu + λ|u|^αu \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $λ\in {\mathbb C}$, $\Re λ<0$ and $0<α<\frac2N$. We give a precise description of the behavior of the solutions (including decay rates in $L^2$ and $L^\infty $, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that $α$ is sufficiently close to $\frac2N$.
The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schrödinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded positive solution at zero energy and are a nowhere dense set of potentials. We relax this assumption to include logarithmically growing positive solutions at zero energy. These potentials are stable under perturbations. Assuming only that the potential is subcritical and has two weak derivatives in a weighted Sobolev space, we prove that the associated scatteri…
This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrödinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted $L^2$ space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previou…
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].
For the 3D focusing cubic nonlinear Schrodinger equation, Scattering of $H^1$ solutions inside the (scale invariant) potential well was established by Holmer and Roudenko~\cite{HR2} (radial case) and Duyckaerts, Holmer and Roudenko~\cite{DHR} (general case). In this paper, we extend this result to arbitrary space dimensions and focusing, mass-supercritical and energy-subcritical power nonlinearities, by adapting the method of~\cite{DHR}.
The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the scattering results are known. In this paper we choose $15/16<s<1$ and assume that the nonlinearity equals $-|u|^p u$ with $p = 2/{3/2 -s}$, so that this problem is $\dot{H}^s \times \dot{H}^{s-1}$ critical. We will prove the global well-posedness and scattering of the solution under the additional assumption that the $\dot{H}^s \times \dot{H}^{s-1}$ norm of the sol…
Initial-boundary value problem for the generalized Zakharov-Kuznetsov equation posed on a bounded rectangle is considered. Critical and subcritical powers in nonlinearity are studied.
Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity
针对一维散焦NLS次临界长程非线性，通过线性化与相位修正构造修正波算子，证明无大小限制下全局解散射到给定渐近轮廓。
In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-Δ)^αu=g(u)+σν\quad{\rm in}\quad Ω,$$ subject to the condition (E2) $$u=\varrhoμ\quad {\rm on}\quad \partialΩ \ {\rm if}\ α=1\qquad {\rm or\ \ in}\ \ Ω^c \ \ {\rm if}\ α\in(0,1),$$ where $σ,\varrho\ge0$, $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(0,1)$ or Laplacian operator if $α=1$, $ν,μ$ are suitable Radon measures and $g:\mathbb{R}_+\mapsto\mathbb{R}_+$ is a continuous function. We introduce an approach to obtain weak solutions for probl…
We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by establishing the refined analysis of the spatially decaying nonlinear term.
We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-itΔ} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.
Construction of multi-solitary waves solution to the focusing nonlinear Schrödinger equation outside an obstacle in the $L^2$-subcritical case
本文在L^2次临界情形下，针对障碍物外部的聚焦非线性薛定谔方程，利用紧性论证、调制理论与局部能量估计，构造了渐近表现为多个不同速度孤立波之和且满足Dirichlet边界条件的解。
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω\subset\R^N$ with Dirichlet boundary conditions on $\partialΩ$. The operator $L$ is a uniformly elliptic linear operator of order $2m$ whose principle part is of the form $\big(-\sum_{i,j=1}^N a_{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j}\big)^m$. We assume that $f$ is superlinear at the origin and satisfies $\lim \limits_{s\to\infty}\frac{f(x,s)}{s^q}=h(x)$, $\lim \limits_{s\to-\infty}\frac{f(x,s)}{|s|^q}=k(x)$, where $h,k\in C(\overlineΩ)$ are positive functions and $q>1$ is subcrit…
We consider ground states of $L^2$-subcritical nonlinear Schrödinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem $$ e(ρ):=\inf\{E_ρ(u):u\in \mathcal{H}(\mathbb{R}^d),\|u\|_2^2=1\}.$$ The energy functional $E_ρ(u)$ is defined by $$ E_ρ(u):=\frac{1}{2}\int_{\mathbb{R}^d}|\nabla u|^2dx +\frac{1}{2}\int_{\mathbb{R}^d}V(x)|u|^2dx-\frac{ρ^{p-1}}{p+1}\int_{\mathbb{R}^d}|u|^{p+1}dx,$$ where $d\geq1$, $ρ>0$, $p\in\big(1, 1+\frac{4}{d}\big)$ and $0\leq V(x)\to\infty$ as $|x| \to\infty$. We present a detailed analysis on the concentra…
Asymptotic profiles for the Cauchy problem of semilinear beam equation with two variable coefficients in the subcritical case
研究次临界非线性情形下含两个变系数的半线性梁方程柯西问题解的渐近轮廓，证明其由关联非线性抛物方程的自相似解描述，并利用加权空间中的能量估计完成证明。
The main result of the paper is a global asymptotic stability result for solutions to the Lifschitz-Slyozov-Wagner (LSW) system of equations. This extends some local asymptotic stability results of Niethammer-Velázquez (2006). The method of proof is along similar lines to the one used in a previous paper of the authors. This previous paper proves global asymptotic stability for a class of infinite dimensional dynamical systems for which no Lyapounov function is (apparently) available.
This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-Δ)^{s_1}_{p}u+(-Δ)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb R^N, \end{array} \end{equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential …
We show that the subcritical $d$-dimensional nonlinear Schrödinger equation $i ψ_t + Δψ+ |ψ|^{2 σ} ψ= 0$, where $1<σd<2$, admits smooth solutions that become singular in~$L^p$ for $p^*<p \le \infty$, where $p^*:=\frac{σd}{σd -1}$. Since $\lim_{σd \to 2-} p^* = 2$, these solutions can collapse at any $2<p \le \infty$, and in particular for $p = 2 σ+2$.
Well-posedness in $L^2$ for nonlocal NLS in the full subcritical range of symbol growth
本文研究了一类非局部非线性Schrödinger方程在直线上的适定性，其非线性由具有亚临界高频增长的傅里叶乘子给出，证明了在L^2(R)中全尺度亚临界范围内的局部适定性。
We look for ground state solutions to the Schrödinger-type system \[ \begin{cases} -Δu_j + λ_j u_j = \partial_jF(u)\\ \int_{\rn} u_j^2 \, dx = a_j^2\\ (λ_j,u_j) \in \mathbb{R} \times H^1(\mathbb{R}^N) \end{cases} j \in \{1,\dots,M\} \] with $N,M\ge1$, where $a=(a_1,\dots,a_M) \in ]0,\infty[^M$ is prescribed and $(λ,u) = (λ_1,\dots,λ_M,u_1,\dots u_M)$ is the unknown. We provide generic assumptions about the nonlinearity $F$ which correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is bounded from below for all or some values of $a$. Making use of a recent idea, w…
We show robustness of various truncated and subcritical approximations to the stochastic defocusing mass-critical nonlinear Schrödinger equation (NLS) in dimension $d=1$, whose solution was constructed in [FX18] with one particular such approximation. The key ingredient in the proof is a uniform bound of the solutions to the family of deterministic mass-subcritical defocusing NLS.
Let $Ω$ be a bounded domain in $\mathbb{R}^N$. In this paper, we consider the following nonlinear elliptic equation of $N$-Laplacian type: $-Δ_{N}u=f(x,u)$ where $u\in W_{0}^{1,2}\{0}$ when $f$ is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to the above equation without the Ambrosetti-Rabinowitz $(AR)$ condition. Earlier works in the literature on the existence of nontrivial solutions to $N-$Laplacian in $\mathbb{R}^{N}$ when the nonlinear term $f$ has …
In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -Δu+u=|u|^{r-2}u &\text{in} \; Ω,\\ \\ \frac{\partial u}{\partial ν}=|u|^{q-2}u &\text{on}\;\partialΩ, \end{array} \right. $$ where $Ω\subset \mathbb R^N, N\geq 3$ is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both $q$ and $r$ are subcritical; (ii) $r$ is critical and $q$ is subcritical; (iii) $r$ is subcritical and $q$ is critical; (iv) Both …
This paper considers ground states of mass subcritical rotational nonlinear Schrödinger equation \begin{equation*} -Δu+V(x)u+iΩ(x^\perp\cdot\nabla u)=μu+ρ^{p-1}|u|^{p-1}u \,\ \text{in} \,\ \mathbb{R}^2, \end{equation*} where $V(x)$ is an external potential, $Ω>0$ characterizes the rotational velocity of the trap $V(x)$, $1<p<3$ and $ρ>0$ describes the strength of the attractive interactions. It is shown that ground states of the above equation can be described equivalently by minimizers of the $L^2-$ constrained variational problem. We prove that minimizers exist for any $ρ\in(0,\infty)$ when …
In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schrödinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into $L^p$ setting: 1. Large data local well-posedness for subcritical nonlinearities, 2. Small data global well-posedness for critical nonlinearities, 3. Large data global well-posedness if the subcritical nonlinearity is given by $|u|^{α-1}u$.
This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on …
Global existence and finite-time blow-up for a strongly damped wave--MGT system with fully subcritical logarithmic nonlinearity
研究强阻尼波-MGT系统在全次临界对数非线性下的动力学，通过增广变量和势阱结构，证明稳定集内全局指数衰减、不稳定集内有限时间爆破。
In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -Δu+λ_1u=μ_1 |u|^{p-2}u+βr_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -Δv+λ_2v=μ_2 |v|^{q-2}v+βr_2|u|^{r_1}|v|^{r_2-2}v\quad&\hbox{in}\;\mathbb R^N,\\ \int_{\mathbb R^N}u^2=a_1^2\quad\hbox{and}\;\int_{\mathbb R^N}v^2=a_2^2, \end{cases}$$ where $p,q,r_1+r_2$ can be Sobolev critical. To this purpose, we study the geometry of the Pohozaev manifold and the associated minimizition problem. Under some assumption on $a_1,a_2$ and $β$, we obtain the existence of the positive n…
We study the scattering problem for the nonlinear Schrödinger equation $i\partial_t u + Δu = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $Σ$ holds and the wave operator is well-defined on $Σ$. We show that there exists $0<β<p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+β}$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in t…
We study small data scattering of solutions to Nonlinear Klein-Gordon equations with suitable pure power nonlinearities, posed on $\mathbb{R}^d\times \mathcal{M}^k$ with $k\leq2$ and $d\geq1$ and $\mathcal{M}^k$ a compact Riemannian manifold. As a special case we cover the $H^1-$critical NLKG on $\mathbb{R} \times M^2$.
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We …
We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump Lèvy processes. If the index of the operator $σ$ is in $ (1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a Hölder regularity, and an interior $\rm C^{1,α}$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index …
Asymptotic behavior for the damped Schrödinger equation with nonlinear dissipation
研究质量次临界阻尼薛定谔方程，证明非线性解的最优$L^2$衰减率与线性解一致，并给出任意幂次散射的最优收敛率。
Mass-subcritical Half-Wave Equation with mixed nonlinearities: existence and non-existence of ground states
本文研究带散焦质量次临界扰动的聚焦质量次临界半波方程，证明存在临界质量决定约束极小元的存在性，并在一维情形下证明基态轨道稳定。
Critical quasilinear elliptic systems with convex subcritical perturbations
本文研究由p-拉普拉斯算子驱动的临界拟线性椭圆系统，通过变分方法和非线性特征值理论，证明了在适当维数下对任意参数存在非平凡解，并利用抽象扰动定理处理非齐次情形得到两个解。
We study the incompressible Navier-Stokes equations with a fractional Laplacian and prove the existence of discontinuous Leray-Hopf solutions in the largest critical space with arbitrarily small initial data.
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case is even new in the classical $p$-Laplacian case.
We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate …
In this paper, we consider the nonlinear Schrödinger equation, $$ i\partial_{t}u+Δu= μ|u|^p u, \quad (t,x)\in \mathbb{R}^{d+1}, $$ with $μ=\pm1, p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in (0,\frac4d)$. We prove that under some restrictions on $d,p$, any radial initial data in the critical space $\dot H^{s_c}(\mathbb{R}^d)$ with compact support, implies global well-posedness.
We investigate the existence of ground states with prescribed mass for the NLS energy with combined $L^2$-critical and subcritical nonlinearities, on a general non-compact metric graph $\mathcal{G}$. The interplay between the different nonlinearities creates new phenomena with respect to purely critical or subcritical problems on graphs; from a different perspective, topological and metric properties of the underlying graph drastically influence existence and non-existence of ground states with respect to the analogue problem on the real line.
We consider a perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, and for a certain range of powers, the existence of a ground state solitary wave, which is then shown to be the previously constructed solution. Finally, we pres…
We consider short-range mass-subcritical nonlinear Schrödinger equations and we show that the corresponding solutions with initial data in $Σ$ scatter in $H^1$. Hence we up-grade the classical scattering result proved by Yajima and Tsutsumifrom $L^2$ to $H^1$.We also provide some partial results concerning the scattering of the first order moments, as well as a short proof via lens transform of a classical result due to Tsutsumi and Cazenave-Weissler on the scattering in $Σ$.
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