On the existence of ground states for nonlinear Schrödinger-Poisson equation

This paper is concerned with the existence of ground states for the Schrödinger-Poisson equation , where V(u) is a Hartree type nonlinearity, stemming from the coupling with the Poisson equation, which includes the so-called doping profile or impurities. By means of variational methods in the energy space we show that ground states exist and belong to the Schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value, it is also shown that for values of the total charge greater than this critical value, energy is not bounded from below. In addition, we show that this critical value is the total charge given by the impurities.     

On the existence of integrable solutions for a nonlinear quadratic integral equation

In this paper, we investigate the existence of integrable solution to a nonlinear integral equation, which includes many important integral and functional equations that arise in nonlinear analysis and its applications. Our results are obtained under rather general assumptions. In particular, the solvability of the well known Chandrasekhar equation is discussed under appropriate assumptions. Our analysis uses the technique of measures of weak noncompactness and rely on a variant of Schauder’s fixed point theorem.     

On the nonlinear stability and the existence of selective decay states of 3D quasi-geostrophic potential vorticity equation

In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.     

On the initial layer and the existence theorem for the nonlinear Boltzmann equation

The truncated Hilbert expansion including the initial layer terms is considered. This enables us to replace the singulary perturbed Boltzmann equation by a weakly nonlinear equation. In this way the existence of a strong solution of the Boltzmann equation is obtained for initial data close enough to a local Maxwellian. The solution exists in the physically significant time interval on which smooth solutions to the Euler equations exist.     

On existence and asymptotic stability of solutions of a nonlinear integral equation

We prove an existence theorem for a nonlinear integral equation being a Volterra counterpart of an integral equation arising in the traffic theory. The method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question.     

On the existence of full dimensional KAM torus for fractional nonlinear Schrodinger equation

In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition $$ \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1) $$ where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$ which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation.     

The existence problem for a nonlinear Abel equation on the half-line

We discuss a nonlinear Abel equation on the half-line (−∞,c), c>0. The basic results provide criteria for the existence of nontrivial everywhere positive solutions. They are expressed in terms of the generalized Osgood condition.     

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.     

Global existence and blow-up solutions for a nonlinear shallow water equation

Considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by Dullin, Gottward and Holm. Blow-up can occur only in the form of wave-breaking. A wave-breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow-up rate is established. The blow-up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined.     

On a stationary transport equation

Summary Let Ω, Γ,v, a andX be as described at the beginning of the introduction below, letp∈]1, +∞[, and setq=p/(p-1). Ifp>2, we also assume that the mean curvature {itx}{su(itx)} of Γ is everywhere nonnegative. In this paper we solve the existence problem in spacesX, for equation (1.1) below, ifX=W ₀ ^1,q , orX=W ⁻¹,p. As a by-product, the solvability of (1.1) in spacesW ^1,pandL ^pfollows (without any assumption on {itx}{su(itx)}). For more general results on the above problem, see ref. [1].     

On the existence of a generalized solution to the first initial-boundary value problem for a nonlinear parabolic equation

Summary In the paper first the existence of a classical solution to an initial-boundary value problem for the nonlinear parabolic equation is proved under the standard condition on H?lder continuity of f but a quite general condition on the growth of f. Then, by using the possibility of the approximation of a continuous function by means of H?lder continuous functions, the foregoing result is applied to the proof of the existence of a generalized solution to the first boundary value problem for the same equation where only continuity of f and a weak assumption on the growth of f is required. Entrata in Redazione il 14 giugno 1970.     

Solutions to a stationary nonlinear Black-Scholes type equation

We study by topological methods a nonlinear differential equation generalizing the Black-Scholes formula for an option pricing model with stochastic volatility. We prove the existence of at least a solution of the stationary Dirichlet problem applying an upper and lower solutions method. Moreover, we construct a solution by an iterative procedure.