Quasilinear Term Research Articles

On sign-changing solutions for quasilinear Schrödinger-Poisson system with critical growth

In this paper, we consider the following quasilinear Schrödinger-Poisson system with critical growth: where , is a smooth potential function and g is a appropriate nonlinear function. For the sake of overcoming the technical difficulties caused by the quasilinear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. Moreover, when g satisfying suitable assumptions and sufficiently large μ, we take advantage of constraint variational method, the quantitative deformation lemma, Moser iteration and approximation technique to obtain a least-energy sign-changing solution , which has precisely two nodal domains.

A splitting method to solve a single nonlinear equation with derivative-free iterative schemes

In the paper, we convert a single nonlinear equation to a system consisting of two equations. While a quasi-linear term is added on the first equation, the nonlinear term in the second equation is decomposed at two sides through a weight parameter. After performing a linearization, an iterative scheme is derived, which is proven of third-order convergence for certain parameters. An affine quasi-linear transformation in the plane is established, and the condition for the spectral radius being smaller than one for the convergence of the iterative scheme is derived. By using the splitting method, we can further identify a sufficient condition for the convergence of the iterative scheme. Then, we develop a step-wisely quasi-linear transformation technique to solve nonlinear equations. Proper values of the parameters are qualified by the derived inequalities for both iterative schemes, which accelerate the convergence speed. The performances of the proposed iterative schemes are assessed by numerical tests, whose advantages are fast convergence, saving the function evaluation per iteration and without needing the differential of the given function.

Ground state solutions for quasilinear scalar field equations arising in nonlinear optics

In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space (Hirata et al. in Topol Methods Nonlinear Anal 35:253–276, 2010; Jeanjean in Nonlinear Anal Theory Methods Appl 28:1633–1659, 1997), we prove the existence of a non-trivial weak solution for general nonlinear terms of Berestycki–Lions’ type. The existence of a radial ground state solution and a ground state solution is also established under stronger assumptions on the quasilinear term.

A Unified Approach to Singularly PerturbedQuasilinear Schr\xf6dinger Equations

In this paper we present a unified approach toinvestigate existence and concentration of positive solutions for the following class of quasilinear Schrödingerequations,-ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,\\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}$$-\\varepsilon^2\\Deltau+V(x)u\\mp\\varepsilon^{2+\\gamma}u\\Delta u^2=h(u),\\ \\ x\\in\\mathbb{R}^N,$$\\end{document}where Ngeqslant3, varepsilon > 0, V(x) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter gamma>0. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution u_{varepsilon,gamma} concentrating, as varepsilonrightarrow 0, around minima points of the potential.

Nonclassical stationary and nonstationary problems with weight Neumann conditions for singular equations with KPZ-nonlinearities

ABSTRACT From a unique viewpoint, singular elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms are investigated in cylindrical domains. The weight Neumann condition is set on the lateral area of the cylinder; no condition is set on the base of the cylinder (regardless the type of the equation). Results of two kinds are established: the existence of a limit of each solution (if it exists) along the axis of the cylinder and sufficient conditions of a blow-up (including instant or complete one).

Quasilinear asymptotically periodic Schr\xf6dinger\u2013Poisson system with subcritical growth

The aim of this paper is establishing the existence of a nontrivial solution for the following quasilinear Schrödinger–Poisson system: {−Δu+V(x)u−uΔ(u2)+K(x)ϕ(x)u=g(x,u),x∈R3,−Δϕ=K(x)u2,x∈R3,u∈H1(R3),u>0,\\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}$$ \\left \\{ \\textstyle\\begin{array}{l} -\\Delta u+V(x)u-u\\Delta(u^{2})+K(x)\\phi(x)u=g(x, u),\\quad x\\in\\mathbb {R}^{3}, \\\\ -\\Delta\\phi=K(x)u^{2}, \\quad x\\in\\mathbb{R}^{3},\\\\ u\\in H^{1}(\\mathbb{R}^{3}),\\qquad u>0, \\end{array}\\displaystyle \\right . $$\\end{document} where V, K, g are continuous functions. To overcome the technical difficulties caused by the quasilinear term, we change the variable to guarantee the feasibility of applying the mountain pass theorem to solve the above problems. We use the mountain pass theorem and the concentration–compactness principle as basic tools to gain a nontrivial solution the system possesses under an asymptotic periodicity condition at infinity.

Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction

This paper is concerned with the following quasilinear Schrödinger equation: -\Delta u+V(x)u-\frac{1}{2}\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N, where N\ge 3 , V\in \mathcal{C}(\mathbb R^N,[0,\infty)) and g\in \mathcal{C}(\mathbb{R}, \mathbb{R}) is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under mild assumptions on V and g . Moreover, we establish a minimax characterization of the ground state energy. Especially, we impose some new conditions on V and more general assumptions on g . For this, some new tricks are introduced to overcome the competing effect between the quasilinear term and the superlinear reaction. Hence our results improve and extend recent theorems in several directions.

Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearities

We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or large in $L^{\infty}$, but the spatial derivatives must initially satisfy a nonlinear smallness condition compared to the time derivative. The first main idea of the proof is to construct a quasilinear integrating factor, which allows us to reformulate the wave equation as a system whose solutions remain regular, all the way up to the singularity. This is equivalent to constructing quasilinear vectorfields adapted to the nonlinear flow. The second main idea is to exploit some crucial monotonic terms in various estimates, especially the energy estimates, that feature the integrating factor. The availability of the monotonicity is tied to our assumptions on the data and on the structure of the quasilinear terms. The third main idea is to propagate the relative smallness of the spatial derivatives all the way up to the singularity so that the solution behaves, in many ways, like an ODE solution. As a corollary of our main results, we show that there are quasilinear wave equations that exhibit two distinct kinds of blowup: the formation of shocks for one non-trivial set of data, and ODE-type blowup for another non-trivial set.

Multiple small solutions for Schrödinger equations involving positive quasilinear term

Multiple small solutions for Schrödinger equations involving positive quasilinear term

The existence of sign-changing solution for a class of quasilinear Schr\xf6dinger\u2013Poisson systems via perturbation method

This paper is concerned with the existence of a sign-changing solution to a class of quasilinear Schrödinger–Poisson systems. There are some technical difficulties in applying variational methods directly to the problem because the quasilinear term makes it impossible to find a suitable space in which the corresponding functional possesses both smoothness and compactness properties. In order to overcome the difficulties caused by nonlocal term and quasi-linear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. And then, by using the approximation technique, a sign-changing solution with precisely two nodal domains is derived.

On absence of global solutions of quasilinear differential-convolutional inequalities

ABSTRACTFrom a unique viewpoint, elliptic and parabolic second-order inequalities with quasilinear KPZ-type terms and nonlocal convolutional terms arising in the description of reaction–diffusion processes, neural networks, and nonlocal phase transitions are investigated. Sufficient conditions of the absence of their global solutions, i.e. necessary conditions of their global solvability, are found.

Homogenization of a class of singular elliptic problems in perforated domains

In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by ε-periodic holes of size ε. The quasilinear equations present a nonlinear singular lower order term fζ(uε), where uε is the solution of the problem at ε-level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new.The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of uε behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence {uε}, as ε→0. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem.The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as fζ(uε). This plays an essential role in order to get an almost everywhere convergence of the solution uε, needed in the study the asymptotic behavior of the problem.

An elliptic system with logarithmic nonlinearity

Abstract In the present paper, we study the existence of solutions for some classes of singular systems involving the {\Delta_{p(x)}} and {\Delta_{q(x)}} Laplacian operators. The approach is based on bifurcation theory and the sub-supersolution method for systems of quasilinear equations involving singular terms.

Existence of long time solutions and validity of the nonlinear Schrödinger approximation for a quasilinear dispersive equation

We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order O(ε) in Sobolev norms exist for a time span of order O(ε−2) for sufficiently small ε. Secondly, we derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the NLS equation.The proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal-form transforms. To justify the NLS approximation, we have to overcome additional difficulties caused by the occurrence of resonances. We expect that the method developed in the present paper will also allow to prove the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances.

Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear Klein-Gordon equation. It is the purpose of this paper to present a method which allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein-Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the first validity proof of the NLS approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in Sobolev spaces. We expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the NLS approximation for other quasilinear hyperbolic systems.

Longtime dynamics of the quasi-linear wave equations with structural damping and supercritical nonlinearities**The work is supported by National Natural Science Foundation of China (No.11271336; No.11671367).

The paper investigates the well-posedness and the longtime dynamics of the quasilinear wave equations with structural damping and supercritical nonlinearities: , with . The main results are concerned with the quasilinear term and the nonlinearities g(u) with supercritical growth. Under the rather mild conditions, the well-posedness and the existence of the global and exponential attractors (rather than the weak ones) are established in natural energy space. These results show that even for the supercritical nonlinearities, the regularity and the longtime dynamics of the solutions of the mentioned equations are of the characteristics of the parabolic equations because of the effectiveness of the structural damping.

A Finite-Orbit-Width Fokker-Planck solver for modeling of energetic particle interactions with waves, with application to Helicons in ITER

The bounce-average (BA) finite-difference Fokker-Planck (FP) code CQL3D [1,2] now includes the essential physics to describe the RF heating of Finite-Orbit-Width (FOW) ions in tokamaks. The FP equation is reformulated in terms of Constants-Of-Motion coordinates, which we select to be particle speed, pitch angle, and major radius on the equatorial plane thus obtaining the distribution function directly at this location. Full-orbit, low collisionality neoclassical radial transport emerges from averaging the local friction and diffusion coefficients along guiding center orbits. Similarly, the BA of local quasilinear RF diffusion terms gives rise to additional radial transport. The local RF electric field components needed for the BA operator are usually obtained by a ray-tracing code, such as GENRAY, or in conjunction with full-wave codes. As a new, practical application, the CQL3D-FOW version is used for simulation of alpha-particle heating by high-harmonic waves in ITER. Coupling of high harmonic or helicon fast waves power to electrons is a promising current drive (CD) scenario for high beta plasmas. However, the efficiency of current drive can be diminished by parasitic channeling of RF power into fast ions, such as alphas, through finite Larmor-radius effects. We investigate possibilities to reduce the fast ion heating in CD scenarios.

Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

We consider a strongly damped quasilinear membrane equation with Dirichlet boundary condition. The goal is to prove the well-posedness of the equation in weak and strong senses. By setting suitable function spaces and making use of the properties of the quasilinear term in the equation, we have proved the fundamental results on existence, uniqueness, and continuous dependence on data including bilinear term of weak and strong solutions.

Multiple solutions of a parameter-dependent quasilinear elliptic equation

Consider the quasilinear elliptic equation Open image in new window where \(\Omega \subset \mathbb R^N\) (\(N\ge 2\)) is a bounded domain with smooth boundary and \(\lambda >0\) is a parameter. For the quasilinear term, we assume that \(a_{ij}=a_{ji}\) and \(a_{ij}(x,s)\)’s growth is like \((1+s^{2})\delta _{ij}\). The nonlinearity of power growth \(f(x,s)=|s|^{r-2}s\) with \(2<r<4\) acts as a typical example of the nonlinear term f, a case in which less results are known compared with the cases \(1<r<2\) and \(4<r<4N/(N-2)^+\). We show the structure of solutions depends keenly upon the parameter \(\lambda \). More precisely, while such an equation has no nontrivial solution for \(\lambda \) small, we prove that both the number of solutions with positive energies and the number of solutions with negative energies tend to infinity as \(\lambda \rightarrow +\infty \). Nodal properties are determined for six solutions among all of them.

A fully-neoclassical finite-orbit-width version of the CQL3D Fokker–Planck code

The time-dependent bounce-averaged CQL3D flux-conservative finite-difference Fokker–Planck equation (FPE) solver has been upgraded to include finite-orbit-width (FOW) capabilities which are necessary for an accurate description of neoclassical transport, losses to the walls, and transfer of particles, momentum, and heat to the scrape-off layer. The FOW modifications are implemented in the formulation of the neutral beam source, collision operator, RF quasilinear diffusion operator, and in synthetic particle diagnostics. The collisional neoclassical radial transport appears naturally in the FOW version due to the orbit-averaging of local collision coefficients coupled with transformation coefficients from local (R, Z) coordinates along each guiding-center orbit to the corresponding midplane computational coordinates, where the FPE is solved. In a similar way, the local quasilinear RF diffusion terms give rise to additional radial transport of orbits. We note that the neoclassical results are obtained for ‘full’ orbits, not dependent on a common small orbit-width approximation. Results of validation tests for the FOW version are also presented.