Nonlinear Neumann Boundary Value Problems Research Articles

Blow up of solutions to nonlinear Neumann boundary value problem for Schrödinger equations

Blow up of solutions to nonlinear Neumann boundary value problem for Schrödinger equations

Existence and approximate solutions of a nonlinear model for the Antarctic circumpolar current

We consider a nonlinear Neumann boundary value problem which is derived for the Antarctic Circumpolar Current. By the theory of topological degree, we prove the existence results for the problem with semilinear oceanic vorticity term. We also construct the approximate solutions for such a nonlinear model.

Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂ t u − Δ u = λ | u | p , t > 0 , x ∈ R + n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R + n , − ∂ x u ( t , x ′ , 0 ) = γ | u | q ( t , x ′ , 0 ) , t > 0 , x ′ ∈ R n − 1 where p = 1 + 2 n , q = 1 + 1 n and ε > 0 is small enough. We investigate the life span of solutions for λ , γ > 0. Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ , γ < 0 and ∫ R + n u 0 ( x ) d x > 0.

A representation theorem approach to probabilistic interpretation for viscosity solutions of Isaacs equations

The main objective of this paper is to show a certain representation theorem for generators of generalized backward stochastic differential equations (GBSDEs) and its application to probabilistic interpretation for viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary value problems. In the representation theorem a time change method is adopted to treat the difficulty brought from the random measure contained in GBSDEs. By means of the representation theorem the viscosity solution of Isaacs equations with nonlinear Neumann problems is proved to be the value function of a two-player zero-sum stochastic differential game problem with state constraints and recursive cost functionals.

Existence of solutions for a class of nonlinear Neumann boundary value problems in the presence of upper and lower solutions

In this article, we explore the monotone iterative technique (MI‐technique) to study the existence of solutions for a class of nonlinear Neumann four‐point, boundary value problems (BVPs) defined as, where and . The nonlinear term , where , is Lipschitz in and one‐sided Lipschitz in . Using lower solution and upper solutions , we develop MI‐technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti‐maximum principle. Then under certain assumptions, we prove that these sequences converge uniformly to the solution in the specific region, where or . To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi‐point BVPs. We don't require sign restriction on which is very common and strict condition.

Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem where k 1 and k 2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × ℝ+, ℝ), ℝ+ := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

Weak Solutions for Nonlinear Neumann Boundary Value Problems with $p(x)$-Laplacian Operators

We study the nonlinear Neumann boundary value problem with a $p(x)$-Laplacian operator \[ \begin{cases} \Delta_{p(x)}u + a(x)|u|^{p(x)-2}u = f(x,u) &\textrm{in $\Omega$}, |\nabla u|^{p(x)-2} \dfrac{\partial u}{\partial\nu} = |u|^{q(x)-2}u + \lambda |u|^{w(x)-2}u &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^N$, with $N \geq 2$, is a bounded domain with smooth boundary and $q(x)$ is critical in the context of variable exponent $p_*(x) = (N-1)p(x)/(N-p(x))$. Using the variational method and a version of the concentration-compactness principle for the Sobolev trace immersion with variable exponents, we establish the existence and multiplicity of weak solutions for the above problem.

Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operatorwhere the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type

We discuss the existence of weak solutions to one class of Neumann boundary value problems (BVP) for non-linear elliptic equations. We introduce a special family of perturbed optimal control problems (OCPs) where the class of fictitious controls is closely related with the properties of the distribution in the right-hand side of the elliptic equation, and we show that optimal solutions of such problems allow one to attain (in the limit) some approximate solutions as the parameter of perturbation $\varepsilon>0$ tends to zero. The main questions we discuss in this paper touch on the solvability of perturbed OCPs, the uniqueness of their solutions, the asymptotic properties of optimal pairs as the perturbation parameter $\varepsilon >0$ tends to zero, and deriving of optimality conditions for the perturbed OCPs. As a consequence, we obtain the sufficient conditions of the existence of weak solutions to the given class of non-linear Neumann BVPs and propose a way for their approximation.

Multiplicity results for nonlinear Neumann boundary value problems involving p-Laplace type operators

We consider the existence of at least two or three distinct weak solutions for the nonlinear elliptic equations $$ \textstyle\begin{cases} {-}\operatorname{div}(\varphi(x,\nabla u))+{|u|}^{p-2}u= \lambda f(x,u) &\mbox{in } \Omega, \varphi(x,\nabla u) \frac{\partial u}{\partial n}= \lambda g(x,u) & \mbox{on }\partial\Omega. \end{cases} $$ Here the function $\varphi(x,v)$ is of type $|v|^{p-2}v$ and the functions f, g satisfy a Caratheodory condition. To do this, we give some critical point theorems for continuously differentiable functions with the Cerami condition which are extensions of the recent results in Bonanno (Adv. Nonlinear Anal. 1:205-220, 2012) and Bonanno and Marano (Appl. Anal. 89:1-10, 2010) by applying Zhong’s Ekeland variational principle.

Measure-valued branching processes associated with Neumann nonlinear semiflows

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term −β, which includes the case β(u)=−um, with 0<m<1. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow, induced by the reflected Brownian motion, while the branching occurs on the measures having non-zero traces on the boundary of the open set, with the behavior of the (−β)-superprocess, having as spatial motion the process on the boundary associated to the reflected Brownian motion.

Three solutions for a nonlinear Neumann boundary value problem

Three solutions for a nonlinear Neumann boundary value problem

The probabilistic Brosamler formula for some nonlinear Neumann boundary value problems governed by elliptic possibly degenerate operators

This paper concerns with boundary value problems as { L u+a0u = f in Ω, +c0|u|m−1u = g on ∂Ω, where L is an elliptic possibly degenerate second order operator, a0, c0 are positive function, γ is an oblique exterior vector and m 1 . By means of some arguments close to the Dynamics Programming we prove that the viscosity solution admits a representation formula that can be considered as an extension of probabilistic Brosamler formula of linear Neumann boundary value problems governed by uniformly elliptic operators. Although other generalizations are possible, by simplicity we limit this contribution to the presence of nonlinear terms exclusively on the boundary of the domain. We emphasize that any uniforme ellipticity assumption is required in the paper.

Multiple positive solutions of nonlinear Neumann problems with time and space singularities

Allowing the nonlinear term to be singular with respect to both the time and space variables, we consider the positive solutions of a nonlinear Neumann boundary value problem. By constructing two height functions and estimating the integrations of these height functions, the existence and multiplicity of positive solutions are established.

Weak and entropy solutions to nonlinear Neumann boundary-value problems with variable exponents

In this article, we study the following nonlinear Neumann boundary-value problem − div a(x, ∇u) + |u| p(x)−2 u = f in Ω, on ∂Ω, where Ω is a smooth bounded open domain in ℝ N , is the outer unit normal derivative on ∂Ω, div a(x, ∇u) a p(x)-Laplace type operator. We prove the existence and uniqueness of a weak solution for f ∈ L (p −)′(Ω), the existence and uniqueness of an entropy solution for L 1-data f independent of u and the existence of weak solutions for f dependent on u. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Existence of solutions to nonlinear Neumann boundary value problems with p-Laplacian operator and iterative construction

By using some results of pseudo-monotone operator, we discuss the existence and uniqueness of the solution of one kind nonlinear Neumann boundary value problems involving the p-Laplacian operator. We also construct an iterative scheme converging strongly to this solution.

Nonexistence, existence and multiplicity of positive solutions to the [formula omitted]-Laplacian nonlinear Neumann boundary value problem

We study the nonexistence, existence and multiplicity of positive solutions for the nonlinear Neumann boundary value problem involving the p ( x ) -Laplacian of the form { − Δ p ( x ) u + λ | u | p ( x ) − 2 u = f ( x , u ) in Ω | ∇ u | p ( x ) − 2 ∂ u ∂ η = g ( x , u ) on ∂ Ω , where Ω is a bounded smooth domain in R N , p ∈ C 1 ( Ω ¯ ) and p ( x ) > 1 for x ∈ Ω ¯ . Using the sub–supersolution method and the variational principles, under appropriate assumptions on f and g , we prove that there exists λ ∗ > 0 such that the problem has at least two positive solutions if λ > λ ∗ , has at least one positive solution if λ = λ ∗ and has no positive solution if λ < λ ∗ .

Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces

AbstractIn this paper we study the existence of radial solutions for Neumann problems in a ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces. Our approach relies on the Leray‐Schauder degree together with some fixed point reformulations of our nonlinear Neumann boundary value problems (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Three Solutions for a Discrete Nonlinear Neumann Problem Involving the p-Laplacian

We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the -Laplacian. Our approach is based on three critical points theorems.

Про існування Н1 -роз'вязків для деяких задач реєстрації зображень

The solubility of the class of nonlinear optimization problems arising in image regis­tration is discussed. The necessary optimility conditions (Euler-Lagrange equation) for such kind of problems is a nonlinear Neumann boundary value problem which is not known to have a solution in general. However, in the image registration context some assumptions can be made that let us move a little bit further in this question.