Concentration-compactness Method Research Articles

On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup

<p>This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.</p>

Infinitely many solutions for p-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method

We prove the existence of infinitely many solutions to a fractional Choquard type equation(−Δ)psu+V(x)|u|p−2u=(K⁎g(u))g′(u)+εWW(x)f′(u)in RN involving the fractional p-Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives sequences of critical values converging to zero for even functionals. To assure the (PS)c conditions, we also use a nonlocal version of the concentration compactness lemma.

The radial bi-harmonic generalized Hartree equation revisited

This note studies the fourth-order generalized Hartree equation$ i\dot u+\Delta^2 u\pm|u|^{p-2}(J_\gamma*|u|^p)u = 0,\quad p\geq2. $Indeed, for both attractive and repulsive sign, the scattering is obtained in the inter-critical regime, which is given by $ 0<s_c<2 $, where the critical Sobolev exponent is given by the equality $ \kappa^\frac{4+\gamma}{2(p-1)}\|u(\kappa^4\cdot,\kappa t)\|_{\dot H^{s_c}} = \|u(\kappa t)\|_{\dot H^{s_c}} $. In the focusing sign, the scattering follows the method due to Dodson and Murphy (Proc. Am. Math. Soc., 145, no. 11 (2017), 4859-4867). This approach is based on a scattering criteria and a Morawetz estimate. This avoids the concentration-compactness method which requires a heavy machinery in order to obtain the desired space-time bounds. The Kenig-Merle road-map was used by the first author, in a previous paper, in order to obtain the scattering. One assumes here that the data is spherically symmetric. This condition will be removed in a paper in progress. In the defocusing regime, the scattering is based on the decay of solutions in some Lebesgue norms coupled with a Morawetz estimate. In order to prove the Morawetz estimates, one assume that the space dimension is $ N\geq5 $. Moreover, one assumes that $ p\geq2 $ in order to avoid a singularity of the source term. The energy scattering implies that the energy global solutions to the considered equation are asymptotic to $ e^{i\cdot\Delta^2}u_\pm $, when $ t\to\pm\infty $. This means that the source term has no effect for large time.

Standing waves for semilinear Schrödinger equations with discontinuous dispersion

We consider here semilinear Schrödinger equations with a non standard dispersion that is discontinuous at $$x=0$$ . We first establish the existence and uniqueness of standing wave solutions for these equations. We then study the orbital stability of these standing wave into a subspace of the energy space that where classical methods as the concentration-compactness method of P.L. Lions can be used.

Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six

This paper deals with the Cauchy problem associated with a nonlinear system of Schrödinger equations with general quadratic-type nonlinearities. The main interest is in proving the existence of blow-up solutions in dimensions five and six. We give sufficient conditions for the existence of such solutions based on the mass and the energy of the associated ground states. The existence of ground states in dimension five was already obtained in a previous paper. In the present manuscript we also establish the existence of such a special solutions in dimension six. This result can also be viewed as of independent interest. The technique we use is based on the concentration-compactness method. The blow-up solutions are obtained without the mass-resonance condition, when the initial data is radial.

A new proof of scattering for the 5D radial focusing Hartree equation

We revisit the scattering result below the ground state of Y. Gao and H. Wu [Scattering for the focusing -critical Hartree equation in energy space. Nonlinear Anal. 2010;73:1043–1056] on the radial focusing energy-subcritical Hartree equation in d = 5, using the method in Dodson and Murphy [A new proof of scattering below the ground state for the 3D radial focusing cubic NLS. preprint, arXiv:1712.09962]. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin instead of using the concentration compactness method in Gao and Wu [Scattering for the focusing -critical Hartree equation in energy space. Nonlinear Anal. 2010;73:1043–1056].

Positive Solutions and Infinitely Many Solutions for a Weakly Coupled System

We study a Schrodinger system with the sum of linear and nonlinear couplings. Applying index theory, we obtain infinitely many solutions for the system with periodic potentials. Moreover, by using the concentration compactness method, we prove the existence and nonexistence of ground state solutions for the system with close-to-periodic potentials.

A system of Schrödinger equations with general quadratic-type nonlinearities

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 of 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.

New Results About the Lambda Constant and Ground States of the 𝑊-Functional

Abstract In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold ( M , g ) {(M,g)} . One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold ( M , g ) {(M,g)} , if the scalar curvature s of ( M , g ) {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s = 0 {s=0} in M. We also introduce a new constant d ⁢ ( M , g ) {d(M,g)} . For the existence of the ground states, we use Lions’ concentration-compactness method.

Ground state solution for a class of Schrödinger–Poisson-type systems with partial potential

In this paper, we consider the existence of positive ground state solution for a class of Schrodinger–Poisson-type systems with both nonlinear critical growth and nonlocal critical growth when the coefficient of the potential function is neither coercive nor periodic and asymptotic periodic. The usual concentration–compactness method is invalid; hence, we employ the modified concentration–compactness principle and mountain pass theorem to deal with the system.

Normalized multi-bump solutions for saturable Schrödinger equations

AbstractIn this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function:$$\begin{array}{} \displaystyle -{\it\Delta} v +{\it\Gamma} \frac{I(\varepsilon x) + v^2}{1+I(\varepsilon x) +v^2} v =\lambda v,\, x\in{{\mathbb{R}}^{2}}. \end{array}$$We prove that, with the density function being radially symmetric, for given integerk≥ 2 there exist a family of non-radial,k-bump type normalized solutions (i.e., with theL2constraint) which concentrate at the global maximum points of density functions whenε→ 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.

On the existence of skyrmions in planar liquid crystals

The study of topologically nontrivial field configurations is an important topic in many branches of physics and applied sciences. In this paper we are interested to the existence of such structures, the so-called skyrmions, in the context of liquid crystals. More precisely, we consider a two-dimensional nematic or cholesteric liquid crystal. In the nematic case we use a Bogomol'nyi type decomposition in order to get a topological lower bound on the configurations with a given degree for the full Oseen-Frank energy functional, and so we can find a global minimum of degree $\pm 1$ for the energy. Then we consider the cholesteric case in presence of an electric field under the one constant approximation assumption, and, by using the concentration-compactness method, we prove the existence of a minimum again on the configurations of degree $\pm 1$, for sufficiently large electric fields.

Existence of solutions for critical Choquard equations via the concentration-compactness method

AbstractIn this paper, we consider the nonlinear Choquard equation$$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$where 0 < μ <N,N⩾ 3,g(u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality and$G(u)=\int ^u_0g(s)\,{\rm d}s$. Firstly, by assuming that the potentialV(x) might be sign-changing, we study the existence of Mountain-Pass solution via a nonlocal version of the second concentration- compactness principle. Secondly, under the conditions introduced by Benci and Cerami , we also study the existence of high energy solution by using a nonlocal version of global compactness lemma.

Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in [formula omitted

In this paper, we study the existence and concentration behavior of semiclassical ground states for a class of saturable nonlinear Schrödinger equations with intensity functions in R2: (0.1)−ε2Δv+ΓI(x)+v21+I(x)+v2v=λv,forx∈R2.We show that for sufficiently negatively large coupling constant Γ and sufficiently small ε there exists a family of normalized ground states (i.e., with the L2 constraint) of the problem. We prove that the family of solutions concentrate around the maxima of the intensity function as ε→0. Our method is variational and depends upon a convexity technique together with the concentration-compactness method.

Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction

We study the existence and stability of standing waves solutions of a three-coupled nonlinear Schrödinger system related to the Raman amplification in a plasma. By means of the concentration-compactness method, we provide a characterization of the standing waves solutions as minimizers of an energy functional subject to three independent L2 mass constraints. As a consequence, we establish existence and orbital stability of solitary waves.

Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in [formula omitted

Conventionally, the existence and orbital stability of ground states of nonlinear Schrödinger (NLS) equations with power-law nonlinearity (subcritical case) can be proved by an argument using strict subadditivity of the ground state energy and the concentration compactness method of Cazenave and Lions [4]. However, for saturable nonlinearity, such an argument is not applicable because strict subadditivity of the ground state energy fails in this case. Here we use a convexity argument to prove the existence and orbital stability of ground states of NLS equations with saturable nonlinearity and intensity functions in R2. Besides, we derive the energy estimate of ground states of saturable NLS equations with intensity functions using the eigenvalue estimate of saturable NLS equations without intensity function.

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrodinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.

Scattering theory for the defocusing energy-supercritical nonlinear wave equation with a convolution

In this paper, we study the global well-posedness and scattering problem for the nonlinear wave equation with a convolution utt−Δu+(|x|−γ∗|u|2)u=0 in dimensions d≥6. We show that if the solution u is apriorily bounded in the critical homogeneous Sobolev space, that is, u∈Lt∞(Ḣxsc×Ḣxsc−1), with sc≔γ−22>1, then u is global and it scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. Our analysis derived from the concentration compactness method to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios: the finite time blow-up solution, the soliton-like solution and the low-to-high frequency cascade. We note that, authors preclude the finite time blow-up solution to wave equation usually by the property of the finite speed of propagation in the previous literature, however, the finite speed of propagation is broken in the nonlocal nonlinear wave equation with a convolution nonlinearity. For this, we will initially establish the low regularity results for almost periodic solutions including the finite time blow-up solutions and initially preclude the finite time blow-up solutions by lowering the regularity.

Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case

We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in H1(Rd) and then utilize the concentration–compactness method to show the global wellposedness and scattering versus blowup in H1(Rd) below the threshold for radial data when d≤4.

Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity

This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method.