Multiple solutions for Grushin operator without odd nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] is a continuous function does not satisfy the Ambrosetti–Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland’s variational principle, existence of two different solutions for [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.

SOLUTIONS WITH MULTIPLE ALTERNATE SIGN PEAKS ALONG A BOUNDARY GEODESIC TO A SEMILINEAR DIRICHLET PROBLEM

We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problem [Formula: see text] where Ω is a smooth and bounded domain of ℝN, ε is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. In particular we prove that if Ω has a plane of symmetry and its intersection with the plane is a two-dimensional strictly convex domain, then, provided that k is even and sufficiently large, a k-peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of ∂Ω.

Existence of Multiple Nontrivial Solutions for a Strongly Indefinite Schrödinger-Poisson System

We consider a Schrödinger-Poisson system inℝ3with a strongly indefinite potential and a general nonlinearity. Its variational functional does not satisfy the global linking geometry. We obtain a nontrivial solution and, in case of odd nonlinearity, infinitely many solutions using the local linking and improved fountain theorems, respectively.

Existence Results for a Perturbed Problem Involving Fractional Laplacians

We extend the results of Cabre and Sire (2011) to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity(-Δ)su=b(x)f(u)inℝwiths∈(0,1). Herebis a positive periodic perturbation forf, and-fis the derivative of a balanced well potentialG. That is,G∈C2,γsatisfiesG(1)=G(-1)<G(τ) ∀τ∈(-1,1), G'(1)=G'(-1)=0.First, for odd nonlinearityfand for everys∈(0,1), we prove that there exists a layer solution via the monotone iteration method. Besides, existence results are obtained by variational methods fors∈(1/2,1)and for more general nonlinearities. While the cases≤1/2remains open.

Vector Operations in Neural Network Computations

Nonlinearity is an important factor in the biological visual neural networks. Among prominent features of the visual networks, movement detections are carried out in the visual cortex. The visual cortex for the movement detection, consist of two layered networks, called the primary visual cortex (V1), followed by the middle temporal area (MT), in which nonlinear functions will play important roles in the visual systems. These networks will be decomposed to asymmetric sub-networks with nonlinearities. In this paper, the fundamental characteristics in asymmetric and symmetric neural networks with nonlinearities are developed for the detection of the changing stimulus or the movement detection in these neural networks. By the optimization of the asymmetric networks, movement detection Equations are derived. Then, it was clarified that the even – odd nonlinearity combined asymmetric networks, has the ability of generating directional vector in the stimulus change detection or movement detection, while symmetric networks need the time memory to have the same ability. Further, the vector operations in the neural network are developed. These facts are applied to two layered networks, V1 and MT.

Analytical Approximations to Conservative Oscillators With Odd Nonlinearity Using the Variational Iteration Method

In this article, a new technique is introduced for establishing analytical approximate solutions to conservative oscillators with strong odd nonlinearity using the variational iteration method and the Fourier series. The illustrated examples show that only a few iterations can provide very accurate approximate solutions for the whole range of oscillation amplitude even for longer time ranges.