Nonlinear Scalar Field Equations Research Articles

Entire Solutions to Nonlinear Scalar Field Equations with Indefinite Linear Part

Abstract We consider the stationary semilinear Schrödinger equation −Δu + a(x)u = f (x, u), u ∈ H1(ℝN), where a and f are continuous functions converging to some limits a∞ > 0 and f∞ = f∞(u) as |x| → ∞. In the indefinite setting where the Schrödinger operator −Δ + a has negative eigenvalues, we combine a reduction method with a topological argument to prove the existence of a solution of our problem under weak one-sided asymptotic estimates. The minimal energy level need not be attained in this case. In a second part of the paper, we prove the existence of ground-state solutions under more restrictive assumptions on a and f . We stress that for some of our results we also allow zero to lie in the spectrum of −Δ+a.

On radial solutions of inhomogeneous nonlinear scalar field equations

We study the existence of radially symmetric solutions u ∈ H 1 ( Ω ) of the following nonlinear scalar field equation − Δ u = g ( | x | , u ) in Ω. Here Ω = R N or { x ∈ R N | | x | > R } , N ⩾ 2 . We generalize the results of Li and Li (1993) [13] and Li (1990) [14] in which they studied the problem in R N and { | x | > R } with the Dirichlet boundary condition. Furthermore, we extend it to the Neumann boundary problem and we also consider the nonlinear Schrödinger equation that is the case g ( r , s ) = − V ( r ) s + g ˜ ( s ) .

Traveling Wave Exact Solutions for Nonlinear Coupled Scalar Field Equations

In the present paper, with the aid of symbolic computation, the nonlinear coupled scalar field equations relevant to materials physics are investigated by using the trigonometric function transform method. More exact traveling wave solutions are obtained for nonlinear coupled scalar field equations. The solutions obtained in this paper include four kinds of soliton solutions and four kinds of trigonometric function solutions.

Existence of a ground state solution for a nonlinear scalar field equation with critical growth

In the present paper, we establish the existence of Ground State Solutions for some class of Elliptic problems with Critical Growth in \({\mathbb{R}^{N}}\) for N ≥ 2. Our results complete the study made in Berestycki and Lions (Arch Rat Mech Anal 82:313–346, 1983) and Berestycki, Gallouet and Kavian (C R Acad Sci Paris Ser I Math 297:307–310, 1984), in the sense that, in those papers only the subcritical growth was considered.

Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space \({(\mathcal H,\langle\cdot,\cdot\rangle)}\), and interval \({\Lambda\subset(0,+\infty)}\) and a map \({K\in C^2(\mathcal H,\mathbb R)}\) whose gradient is a compact mapping, we consider the family of functionals of the type: $$I(\lambda,u)=\dfrac12\langle u,u\rangle-\lambda K(u),\quad (\lambda,u)\in\Lambda\times\mathcal H.$$ As already observed by many authors, for the functionals we are dealing with the (PS) condition may fail under just this assumptions. Nevertheless, by using a recent deformation Lemma proven by Lucia (Topol Methods Nonlinear Anal 30(1):113–138, 2007), we prove a Poincare–Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for a fourth order nonlinear scalar field equation on a bounded and smooth C∞ region of the four dimensional Euclidean space in the flavor of (Malchiodi in Adv Differ Equ 13:1109–1129, 2008). We remark that this formula has been proven with complete different methods in (Lin and Wei Preprint, 2007) by using blow-up type estimates.

Semilinear equations on fractal blowups

The paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal, where imbedding of the energy space into L p is not compact. In absence of invariant transformations involved in conventional concentration-compactness argument, the paper develops convergence reasoning based on the fractal's self-similarity.

Morse theory and a scalar field equation on compact surfaces

The aim of this paper is to study a nonlinear scalar field equation on a surface $\Sigma$ via a Morse-theoretical approach, based on some of the methods in [25]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [18], which used refined blow-up estimates from [34] and [17]. Related results are derived for the prescribed $Q$-curvature equation on four manifolds.

Asymmetric gravitational lenses in TeVeS and application to the bullet cluster

Aims.We explore the lensing properties of asymmetric matter density distributions in Bekenstein's tensor-vector-scalar theory (TeVeS).

Infinitely many bound states for some nonlinear scalar field equations

In this paper we consider the problem \(-\Delta u + a(x)u = \vert u\vert ^{p-2}u\) in \(\mathbb{R}^N\), where p > 2 and \(p 2. Assuming that the potential a(x) is a regular function such that \(\liminf_{\vert x\vert\rightarrow + \infty} a(x) = a_\infty > 0\) and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.

Multiple Solutions of Nonlinear Scalar Field Equations

We establish existence of multiple solutions of the semilinear elliptic equation where N ≥ 3, 2 < p < 2N/(N − 2), and a(x) and q(x) tend to some positive limits a ∞ and q ∞ respectively as |x| → ∞. Assuming some weak one-sided asymptotic estimates for a(x) and q(x), we prove that (℘) has at least pairs ±u of nontrivial weak solutions.

A remark on least energy solutions in $\mathbf {R}^N$

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in $\mathbf {R}^N$: \begin{equation*} -\Delta u = g(u), u \in H^1(\mathbf {R}^N), \end{equation*} where $N\geq 2$. Without the assumption of the monotonicity of $t\mapsto \frac {g(t)}{t}$, we show that the mountain pass value gives the least energy level.

Threshold of global existence for 3-D attractive nonlinear Schrödinger equations under confined potentials

Based on a nonlinear critical index, a threshold of global existence on the L 2-norm of initial data for 3-D attractive nonlinear Schrödinger equations under confined potentials is derived. For numerical verification, a newly developed radial basis function method is applied in this paper to solve the related nonlinear scalar field equation in R 3.

Exact Solitary Wave Solutions of the Coupled Nonlinear Scalar Field Equations**The project supported partially by National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province of China

Because of the arbitrary property of the singular manifold, the usual singularity analysis can be extended to a different form. Using a nonstandard truncation approach, five special types of exact solution of the coupled nonlinear scalar field equations are obtained.

Dynamics of gravitational waves in 3D: Formulations, methods, and tests

The dynamics of gravitational waves is investigated in full 3+1 dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from non-linearities and gauge degrees of freedom. Using gravitational waves with amplitudes low enough that one has a good understanding of the physics involved, but large enough to enable non-linear effects to emerge, we study the coupling between numerical errors, coordinate effects, and the nonlinearities of the theory. We discuss the various strategies used in identifying specific features of the evolution. We show the importance of the flexibility of being able to use different numerical schemes, different slicing conditions, different formulations of the Einstein equations (standard ADM vs. first order hyperbolic), and different sets of equations (linearized vs. full Einstein equations). A non-linear scalar field equation is presented which captures some properties of the full Einstein equations, and has been useful in our understanding of the coupling between finite differencing errors and non-linearites. We present a set of monitoring devices which have been crucial in our studying of the waves, including Riemann invariants, pseudo-energy momentum tensor, hamiltonian constraint violation, and fourier spectrum analysis.

Spectral asymptotics for nonlinear non-autonomous elliptic eigenvalue problems in a ball

We consider the following nonlinear elliptic eigenvalue problem where is and eigenvalue parameter. We treat this problem in the L2-framework and shall establish an asymptotic formula of variational eignevalue where μα is a positive solution associated with μ(α). The result is connected with the ground state of the nonlinear scalar field equation.

Exact plane symmetric solutions of sin-Gordon equations with inherent gravitational fields taken into account

We obtain exact self-consistent plane symmetric solutions to systems of Einstein equations and nonlinear scalar field equations of sin-Gordon type. We write out, in explicit from, two specific solutions corresponding to specific choices of the nonlinearity parameter λ. We show that for each value of λ, there exist solutions of soliton and antisoliton type with topological charge Q=±1. We establish that the basic difference between the solutions we obtain and solutions of the sin-Gordon equation in flat space—time is that when we take internal gravitational fields into account, each value of λ corresponds to a specific function ϕλ(x).

Nonautonomous Nonlinear Scalar Field Equations in R2

Nonautonomous Nonlinear Scalar Field Equations in R2

Nonlinear scalar field equations

Nonlinear scalar field equations

SEPARATING COORDINATE SYSTEMS IN THE MINKOWSKIAN SPACE-TIME

A class of metrics providing the complete separation of variables in the Klein-Gordon equation is considered. The general exact solution to the vacuum Einstein equation for such metrics is obtained by the variable separation method. It is shown that all these solutions correspond to curvilinear coordinate systems in the Minkowskian space-time. Several limit cases of such systems are investigated. Moreover, some other separating systems are constructed and it is shown that they make it possible to obtain partial exact solutions to nonlinear scalar field equations.

On the Vortex Solutions of Some Nonlinear Scalar Field Equations

On the Vortex Solutions of Some Nonlinear Scalar Field Equations