In this article, we are interested in multi-bump solutions of the singularly perturbed problem \begin{equation*} -ε^2Δv+V(x)v=f(v) \ \ \mbox{ in }\ \ \R^N. \end{equation*} Extending previous results \cite{B, DLY,W1}, we prove the existence of multi-bump solutions for an optimal class of nonlinearities $f$ satisfying the Berestycki-Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as $ε\rightarrow 0$. Examples of potential wells include the following: the union of two compact smooth submanifolds of $\R^N$ where these two submanifolds meet at the origin and an embedded topological submanifold of $\R^N$.
In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems by applying the concentration-compactness principle. Via symmetric decreasing rearrangements, we also show that the standing waves, up to a translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near…
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.
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 establish the existence of multi-bump solutions for the following class of quasilinear problems $$ - Δ_{ p(x) } u + \big( λV(x) + Z(x) \big) u ^{ p(x)-1 } = f(x,u) \text{ in } \mathbb R^N, \, u \ge 0 \text{ in } \mathbb R^N, $$ where the nonlinearity $ f \colon \mathbb R^N \times \mathbb R \to \mathbb R $ is a continuous function having a subcritical growth and potentials $ V, Z \colon \mathbb R^N \to \mathbb R $ are continuous functions verifying some hypotheses. The main tool used is the variational method.
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 are going to study a class of Schrödinger-Poisson system $$ \left\{ \begin{array}{ll} - Δu + (λa(x)+1)u+ ϕu = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -Δϕ=u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $Ω_1, Ω_2, ....., Ω_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.
We investigate the properties of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground states, we establish the orbital stability of standing waves for mass sub-critical nonlinearity. Owing to the non-compactness and the absence of translational invariance of the problem, we apply a profile decomposition argument. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse s…
We consider standing waves with frequency $ω$ for 4-superlinear Schrödinger-Poisson system. For large $ω$ the problem reduces to a system of elliptic equations in $\mathsf{R}^3$ with potential indefinite in sign. The variational functional does not satisfy the mountain pass geometry. The nonlinearity considered here satisfies a condition which is much weaker than the classical (AR) condition and the condition (Je) of Jeanjean. We obtain nontrivial solution and, in case of odd nonlinearity an unbounded sequence of solutions via the local linking theorem and the fountain theorem, respectively.
We investigate the asymptotic stability of standing waves for a model of Schrödinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $δ$) interaction with strength $α$, and letting the strength $α$ depend on the wavefunction in a prescribed way. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $Φ_ω$, and the linearization around them admits two imaginary eigenvalues which in principle could corre…
In this paper, we are going to study the existence of solution for the following Kirchhoff problem $$ \left\{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx +\displaystyle\int_{\mathbb{R}^{3}} λa(x)+1)u^{2} dx\biggl) \biggl(- Δu + (λa(x)+1)u\biggl) = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3}, \\ \mbox{}\\ u \in H^{1}(\mathbb{R}^{3}). \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well with $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $Ω_1, Ω_2, ....., Ω_k$ and the nonlinearity $f(t)$ has a subcritical growth, we ar…
We extend the notion of orbital stability to systems of nonlinear Schrodinger equations, then we prove this property under suitable assumptions of the local nonlinearity involved.
In this paper we study the existence and stability of normalized standing waves for the nonlinear Schrödinger equation on a general starlike graph with potentials. Under general assumptions on the graph and the potential, we show the existence of orbitally stable standing waves when the nonlinearity is $L^{2}$-critical and supercritical.
We study the instability of standing wave solutions for nonlinear Schrödinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.
We study the instability of standing waves for nonlinear Schrödinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will …
We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional "action" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1…
We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit, proving the convergence of bound states.
We study strong instability (instability by blowup) of standing wave solutions for a nonlinear Schrödinger equation with an attractive delta potential and $L^2$-supercritical power nonlinearity in one space dimension. We also compare our sufficient condition on strong instability with some known results on orbital instability.
We study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-Δ)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobo…
In this article, we consider nonlinear Schrödinger equation with nonlocal nonlinearity which is a generalized model of the Schrödinger-Poisson system (Schrödinger-Newton equations) in low dimensions. We first prove the global well-posedness in wider space than in previous result and show the stability of standing waves including excites states. It turns out that an example of stable excite states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.
We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schrödinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact) interaction with strength $α$, which consists of a singular perturbation of the laplacian described by a selfadjoint operator $H_α$, where the strength $α$ depends on the wavefunction: $i\dot u= H_αu$, $α=α(u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, we let the strength $α$ depend on $u$ according to th…
We study the instability of standing-wave solutions $e^{iωt}ϕ_ω(x)$ to the inhomogeneous nonlinear Schrödinger equation $$iϕ_t=-\triangleϕ+|x|^2ϕ-|x|^b|ϕ|^{p-1}ϕ, \qquad \in\mathbb{R}^N, $$ where $ b > 0 $ and $ ϕ_ω $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $ω$ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
We study strong instability of standing waves $e^{iωt} ϕ_ω(x)$ for nonlinear Schrödinger equations with $L^2$-supercritical nonlinearity and a harmonic potential, where $ϕ_ω$ is a ground state of the corresponding stationary problem. We prove that $e^{iωt} ϕ_ω(x)$ is strongly unstable if $\partial_λ^2 E(ϕ_ω^λ) |_{λ=1}\le 0$, where $E$ is the energy and $v^λ(x)=λ^{N/2} v(λx)$ is the $L^2$-invariant scaling.
We study the strong instability of ground-state standing waves $e^{iωt}ϕ_ω(x)$ for $N$-dimensional nonlinear Schrödinger equations with double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the $L^2$-invariant scaling.
We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to p…
We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The perturbed Laplace operator always has a unique simple negative eigenvalue. We prove that if the frequency of the standing wave is close to the negative eigenvalue, it is stable. Moreover, if the frequency is sufficiently large, we have the stability i…
We consider a nonlinear Schrödinger (NLS) equation with any positive power nonlinearity on a star graph $Γ$ ($N$ half-lines glued at the common vertex) with a $δ$ interaction at the vertex. The strength of the interaction is defined by a fixed value $α\in \mathbb{R}$. In the recent works of Adami {\it et al.}, it was shown that for $α\neq 0$ the NLS equation on $Γ$ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $α<0$ that, in the NLS equation with a subcritical power-type nonl…
We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2Δu+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$ of the above problem which concentrates at an isolated component of positive local minimum points of $V$ as $\e\to 0$ under certain conditions on $f$. Our result completes the study made in some very recent works in the sense that, in those papers only the subcritical growth was considered
This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know. In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in O…
In this paper we will continue the analysis of two dimensional Schrödinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the st…
In this paper, the existence and orbital stability of the periodic standing waves solutions for the nonlinear fractional Schrodinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and we it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding positive operators, the oscillation theorem for fractional Hill operators and a Vakhitov-Kolokolov condition, well known for Schrodinger equations. We then perform a numerical app…
In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing wave for the following Chern-Simons-Schrödinger equation \begin{equation} -Δu+u +λ\left(\int_{0}^{\infty}\frac{h(s)}{s}u^{2}(s)ds+\frac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}\right)u=f(x,u)+εk(x)\quad\quad \text{in}\,\,\mathbb{R}^2, \\ \end{equation} where $h(s)=\int_{0}^{s}\frac{l}{2}u^{2}(l)dl$, $λ>0$ and the nonlinearity $f:\mathbb{R}^2\times \mathbb{R}\rightarrow \mathbb{R}$ behaves like $\text{exp}(α\vert u\vert^{2})$ as $\vert u\vert\rightarro…
We are concerned with the following Schrödinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}Δu + V(x)u + ψu = λ|u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill - {\varepsilon ^2}Δψ= {u^2}{\text{in}}{\mathbb{R}^3},{\text{}}u > 0,{\text{}}u \in {H^1}({\mathbb{R}^3}), \hfill \end{gathered} \right. \] where $\varepsilon > 0$ is a small positive parameter, $λ> 0$, $3 < p \le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\varepsilon} \in {H^1}({\mathbb{R}^3})$ which concentrates around a loca…
We study the strong instability of standing waves $e^{iωt}ϕ_ω(x)$ for nonlinear Schrödinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $ω\in\mathbb{R}$ is a frequency, and $ϕ_ω\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta (2018) proved that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, then the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the scaling, which does …
We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.
In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - Δu - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are sh…
In this article, we study the standing-wave solutions to a class of systems of nonlinear Schrödinger equations. Our target is all the standard forms of the NLS systems, with two unknowns, that have a common linear part and cubic gauge-invariant nonlinearities and that yield a Hamiltonian with a coercive kinetic-energy part. We give a necessary and sufficient condition on the existence of the ground state. Further, we give a characterization of the shape of the ground state. It will turn out that the ground states are scalar-type, i.e., multiples of a constant vector and a scalar function. We f…
We study standing waves for a nonlinear Schrödinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $ i \frac{d}{dt}Ψ_t = H Ψ_t - | Ψ_t |^{2μ} Ψ_t $, where $H$ is the Hamiltonian operator which generates the linear Schrödinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $ω> \fra…
This paper is concerned with the Cauchy problem for an inhomogeneous nonlinear Schrodinger equation with exponential growth nonlinearity and harmonic potential in two space dimensions. We prove global well-posedness, existence of the associated ground state and instability of the standing wave.
On a star graph made of $N \geq 3$ halflines (edges) we consider a Schrödinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there exists a family of standing waves, symmetric with respect to the exchange of edges, that can be parametrized by the mass (or $L^2$-norm) of its elements. Furthermore, if the mass is small enough, then the corresponding symmetric standing wave is a ground state and, consequently, it is orbitally stable. On the other hand, if the mass is above a threshold value, then…
In this paper, we study the transverse stability of the line Schrödinger soliton under a full wave guide Schrödinger flow on a cylindrical domain $\mathbb R\times\mathbb T$. When the nonlinearity is of power type $|ψ|^{p-1}ψ$ with $p>1$, we show that there exists a critical frequency $ω_{p} >0$ such that the line standing wave is stable for $0<ω< ω_{p}$ and unstable for $ω> ω_{p}$. Furthermore, we characterize the ground state of the wave guide Schrödinger equation. More precisely, we prove that there exists $ω_{*} \in (0, ω_{p}]$ such that the ground states coincide with the line standing wav…
We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surpris…
We study the nonlinear Schrödinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\in \mathbb{R}$. We investigate an orbital instability of the standing waves $e^{iωt}\mathbfΦ(x)$ of NLS-$δ$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $Φ(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques…
We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.
