Combined Power-type Nonlinearities Research Articles

Normalized ground state of a mixed dispersion nonlinear Schrodinger equation with combined power-type nonlinearities

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.
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Global behavior of the solutions to nonlinear wave equations with combined power-type nonlinearities with variable coefficients

In this paper we study the initial boundary value problem for the nonlinear wave equation with combined power-type nonlinearities with variable coefficients. Existence and uniqueness of local weak solutions are proved. The global behavior of the solutions with non-positive and sub-critical energy is completely investigated. The threshold between global existence and finite time blow up is found. For super-critical energy, two new sufficient conditions guaranteeing blow up of the solutions for a finite time, are given. One of them is proved for an arbitrary sign of the scalar product of the initial data, while the other one is derived only for a positive sign.

Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities

Abstract In this article, we investigate the Cauchy problem for Klein-Gordon equations with combined power-type nonlinearities. Coefficients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coefficients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and sufficient conditions for nonexistence of global solution for subcritical initial energy by means of the sign of the Nehari functional. When the energy is positive, we propose new sufficient conditions for finite time blow up of the weak solutions. One of these conditions is independent of the sign of the scalar product of the initial data. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities.

Normalized ground states of nonlinear biharmonic Schrödinger equations with Sobolev critical growth and combined nonlinearities

This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities Δ2u−λu=μ|u|q−2u+|u|4∗−2uinRN,where N≥5, μ>0, 2<q<2+8N, 4∗=2NN−4 is the H2-critical Sobolev exponent, and λ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are proved to be local minima of the associated energy functional.

Stability and instability of standing waves for the fractional nonlinear Schrödinger equations

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrödinger equation with combined power-type nonlinearities ( FNLS ) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) < 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N < p 2 < 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 < p 1 < p 2 and 4 s N ≤ p 2 < 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 < p 2 < 4 s N − 2 s , or a = 1 , 0 < p 1 < 4 s N < p 2 < 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.

Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

<abstract> In this paper, we investigate the existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities <p class="disp_formula">$ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi = 0,\; \; (t,x)\in [0,T^\star)\times \mathbb{R}^N. $ By using concentration compactness principle, when one nonlinearity is focusing and $ L^2 $-critical, the other is defocusing and $ L^2 $-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper ^{[<xref ref-type="bibr" rid="b14">14</xref>]} to the $ L^2 $-critical and $ L^2 $-supercritical nonlinearities. </abstract>

Sharp thresholds of blow-up and global existence for the Schr\xf6dinger equation with combined power-type and Choquard-type nonlinearities

In this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equationiψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ i\\psi _{t}+\\Delta \\psi =\\lambda _{1} \\vert \\psi \\vert ^{p_{1}}\\psi +\\lambda _{2}\\bigl(I _{\\alpha } \\ast \\vert \\psi \\vert ^{p_{2}}\\bigr) \\vert \\psi \\vert ^{p_{2}-2}\\psi . $$\\end{document} We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the L^{2}-critical case. Our obtained results extend and improve some recent results.

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schr\"odinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

Existence of stable standing waves for the fractional Schrödinger equations with combined power-type and Choquard-type nonlinearities

In this paper, we study the orbital stability of standing waves for the fractional Schrödinger equations with combined power-type and Choquard-type nonlinearities. By using variational methods, when one nonlinearity is focusing and L2-critical, the other is defocusing and L2-supercritical, we prove that there exist the orbitally stable standing waves. We extend the results of Bhattarai [J. Differ. Equations 263, 3197–3229 (2017)] and Feng-Zhang [Comput. Math. Appl. 75, 2499–2507 (2018)] to the L2-critical and L2-supercritical nonlinearities.

On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

We undertake a comprehensive study for the fractional nonlinear Schrodinger equation \begin{document}$ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $\end{document} where \begin{document}$ \frac{d}{2d-1} \leq s , \begin{document}$ 0 . Firstly, we establish the local and global well-posedness results for non-radial and radial \begin{document}$ H^s $\end{document} initial data, radial \begin{document}$ \dot{H}^{s_c}\cap \dot{H}^s $\end{document} initial data, where \begin{document}$ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $\end{document} . Secondly, we study the asymptotic behavior of global radial \begin{document}$ H^s $\end{document} solutions. Of particular interest is the \begin{document}$ L^2 $\end{document} -critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the \begin{document}$ L^2 $\end{document} -critical fractional nonlinear Schrodinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial \begin{document}$ \dot{H}^{s_c} \cap \dot{H}^s $\end{document} solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with \begin{document}$ L^2 $\end{document} -critical and \begin{document}$ L^2 $\end{document} -subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both \begin{document}$ L^2 $\end{document} -critical and \begin{document}$ L^2 $\end{document} -supercritical cases, including mass-concentration and limiting profile.

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where $0<p_1<p_2<\frac{2s}{N-2s}$. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $0<p_1<\frac{2s}{N}$ and $p_2=\frac{2s}{N}$. Finally, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rate and limiting profile.

On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities

This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrodinger equation with combined power-type nonlinearities $$\begin{aligned} iu_{t}+\Delta u=\lambda _1|u|^{p_1}u+\lambda _2|u|^{p_2}u. \end{aligned}$$ When $$p_1=\frac{4}{N}$$ and $$0<p_2<\frac{4}{N}$$ , we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for this equation. This is a complement to the result of Tao et al. (Commun Partial Differ Equ 32:1281–1343, 2007). Moreover, we investigate the dynamical properties of blow-up solutions, including $$L^2$$ -concentration, blow-up rates and limiting profile. When $$\frac{4}{N}<p_1<\frac{4}{N-2}$$ ( $$4<p_1<\infty $$ if $$N=1$$ , $$2<p_1<\infty $$ if $$N=2$$ ), we prove that the blow-up solution with bounded $$\dot{H}^{s_c}$$ -norm must concentrate at least a fixed amount of the $$\dot{H}^{s_c}$$ -norm and, also, its $$L^{p_c}$$ -norm must concentrate at least a fixed $$L^{p_c}$$ -norm.

DIVERGENT SOLUTION TO THE NONLINEAR SCHR&amp;#214;DINGER EQUATION WITH THE COMBINED POWER-TYPE NONLINEARITIES

In this paper, we consider the Cauchy problem for the nonlinear Schr&#246;dinger equation with combined power-type nonlinearities, which is masscritical/supercr-itical, and energy-subcritical. Combing Du, Wu and Zhang' argument with the variational method, we prove that if the energy of the initial data is negative (or under some more general condition), then the <i>H</i>¹-norm of the solution to the Cauchy problem will go to infinity in some finite time or infinite time.

Arbitrary energy global existence for wave equation with combined power-type nonlinearities of different signs

This paper proves the global existence of solution for a class of nonlinear wave equations with nonlinear combined power-type nonlinearities of different signs for the initial data at sup-critical energy level.

Global well-posedness, scattering, and blowup for nonlinear coupled Schrödinger equations in ℝ3

In this paper, we study the system of the coupled nonlinear Schröd-inger equation in dimension three. By establishing the a priori interaction, Morawetz estimate in the spirit of the work of Tao et al. [The nonlinear Schrödinger equation with combined power-type nonlinearities. Commun. Par. Differ. Equ. 2007;32:1281–1343], we prove the global well-posedness and scattering of the solution to the defocusing coupled equation. Furthermore, we adapt the standard method from Cazenave [Semilinear Schrödinger equations. Vol. 10, Courant lecture notes in mathematics. Providence (RI): American Mathematical Society; 2003] to obtain the blowup result for the focusing case under the restriction: energy is negative and the initial data belongs to or the initial data is radial.

Ground States for the Schrödinger Systems with Harmonic Potential and Combined Power-Type Nonlinearities

We consider a class of coupled nonlinear Schrödinger systems with potential terms and combined power-type nonlinearities. We establish the existence of ground states, by using a variational method. As an application, some symmetry results for ground states of Schrödinger systems with harmonic potential terms are obtained.

Global existence to generalized Boussinesq equation with combined power-type nonlinearities

The Cauchy problem to the generalized Boussinesq equation with combined power-type nonlinearities is studied. Global solvability or finite time blow-up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the Nehari functional. For generalized Lienard (or generalized Bernoulli) nonlinear terms the critical energy constant is explicitly evaluated. A new method, that can be considered as a modification of the potential well method, is developed. The performed numerical experiments support the theoretical results.

Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrödinger equations with combined power-type nonlinearities

We study the global existence of solutions in Sobolev spaces and the asymptotic behavior for the Cauchy problem of the following fourth-order Schrödinger equation with combined power-type nonlinearities iut+Δ2u+λ1|u|pu+λ2|u|8n−4u=0, where t∈R, x∈Rn, 8n≤p<8n−4, λ1,λ2 are nonzero real numbers.

Blowing up of solutions to the Cauchy problem for the generalized Zakharov system with combined power-type nonlinearities

This paper deals with blowing up of solutions to the Cauchy problem for a class of generalized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c0 = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α ≥ 0 and \(1 + \tfrac{4} {N} \leqslant p < \tfrac{{N + 2}} {{N - 2}}\) or α < 0 and \(1 < p < 1 + \tfrac{4} {N}\) (N = 2, 3); the other is established under the condition N = 3, \(1 < p < \tfrac{{N + 2}} {{N - 2}}\) and α(p − 3) ≥ 0. On the other hand, for c0 < +∞ and α(p − 3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study.

Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities

In this paper, we study the Cauchy problem of generalized Boussinesq equation with combined power-type nonlinearities utt −uxx + uxxxx + f(u)xx = 0, where or . The arguments powered by potential well method combined with some other analysis skills allow us to give the sharp conditions of global well-posedness. And we also characterize the blow-up phenomenon. Copyright © 2011 John Wiley & Sons, Ltd.