Multiple symmetric nonnegative solutions of second-order ordinary differential equations

AbstractThe existence of multiple nonnegative solutions of the equations −χ″ = f(χ, χ′) subject to χ(0) = χ(1) = 0 is studied. The result is obtained that there are at least three symmetric nonnegative solutions if certain conditions are imposed on f

Existence of positive solutions to Kirchhoff type problems with zero mass

The existence of positive solutions depending on a nonnegative parameter lambda to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais-Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais-Smale sequences for the variable-coefficient case. (C) 2013 Elsevier Inc. All rights reserved

Existence of a positive solution to Kirchhoff type problems without compactness conditions

The existence of a positive solution to a Kirchhoff type problem on R-N is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional is utilized to obtain the bounded Palais-Smale sequences. (c) 2012 Elsevier Inc. All rights reserved

On sign-changing solutions for nonlinear operator equations

AbstractIn this paper, the existence of sign-changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory. The main theorems are some new three-solution theorems which are different from the famous Amann's and Leggett-Williams' three-solution theorems as well as the results in [F. Li, G. Han, Generalization for Amann's and Leggett–Williams' three-solution theorems and applications, J. Math. Anal. Appl. 298 (2004) 638–654]. These three solutions are all nonzero. One of them is positive, another is negative, and the third one is a sign-changing solution. Furthermore, the theoretical results are successfully applied to both integral and differential equations

Towards Self-Interpretable Graph-Level Anomaly Detection

Graph-level anomaly detection (GLAD) aims to identify graphs that exhibit notable dissimilarity compared to the majority in a collection. However, current works primarily focus on evaluating graph-level abnormality while failing to provide meaningful explanations for the predictions, which largely limits their reliability and application scope. In this paper, we investigate a new challenging problem, explainable GLAD, where the learning objective is to predict the abnormality of each graph sample with corresponding explanations, i.e., the vital subgraph that leads to the predictions. To address this challenging problem, we propose a Self-Interpretable Graph aNomaly dETection model (SIGNET for short) that detects anomalous graphs as well as generates informative explanations simultaneously. Specifically, we first introduce the multi-view subgraph information bottleneck (MSIB) framework, serving as the design basis of our self-interpretable GLAD approach. This way SIGNET is able to not only measure the abnormality of each graph based on cross-view mutual information but also provide informative graph rationales by extracting bottleneck subgraphs from the input graph and its dual hypergraph in a self-supervised way. Extensive experiments on 16 datasets demonstrate the anomaly detection capability and self-interpretability of SIGNET.Comment: 23 pages; accepted to NeurIPS 202

Attentional Engagement and Disengagement Differences for Circumscribed Interest Objects in Young Chinese Children with Autism

The current study aimed to investigate attentional processing differences for circumscribed interest (CI) and non-CI objects in young Chinese children with autism spectrum condition (ASC) and typically developing (TD) controls. In Experiment 1, a visual preference task explored attentional allocation to cartoon CI and non-CI materials between the two groups. We found that ASC children (n = 22, 4.95 ± 0.59 years) exhibited a preference for CI-related objects compared to non-CI objects, and this effect was absent in the TD children (n = 22, 5.14 ± 0.44 years). Experiment 2 utilized the traditional gap-overlap paradigm (GOP) to investigate attentional disengagement from CI or non-CI items in both groups (ASC: n = 20, 5.92 ± 1.13 years; TD: n = 25, 5.77 ± 0.77 years). There were no group or stimulus interactions in this study. Experiment 3 adopted a modified GOP (MGOP) to further explore disengagement in the two groups (ASC: n = 20, 5.54 ± 0.95 years; TD: n = 24, 5.75 ± 0.52 years), and the results suggested that exogenous disengagement performance was preserved in the ASC group, but the children with ASC exhibited increased endogenous attentional disengagement compared to TD peers. Moreover, endogenous disengagement was influenced further in the presence of CI-related objects in the ASC children. The current results have implications for understanding how the nature of engagement and disengagement processes can contribute to differences in the development of core cognitive skills in young children with ASC

Ground-state solutions to a class of modified Kirchhoff-type transmissiom problems with critical perturbation

This paper discusses a class of modified Kirchhoff-type transmission problems with critical perturbation. We establish an existence result of the ground-state solutions by using perturbation methods. Meanwhile, the limit properties of solution sequence are investigated

Existence and Iteration of Positive Solutions for Multipoint Boundary Value Problems Dependence on the First Order Derivative with One-Dimensional p-Laplacian

Abstract In this paper, we study the existence of monotone positive solutions for the following nonlinear m-point singular boundary value problem with p-Laplacian operator. The main tool is the monotone iterative technique. We obtain not only the existence of positive solutions for the problem, but also establish iterative schemes for approximating solution. Mathematics Subject Classification: 34B1

Ground-state solutions to a class of modified Kirchhoff-type transmissiom problems with critical perturbation

This paper discusses a class of modified Kirchhoff-type transmissiom problems with critical perturbation. We establish an existence result of the ground-state solutions by using perturbation methods. Meanwhile, the limit properties of solution sequence are investigated

Ground state for Choquard equation with doubly critical growth nonlinearity

In this paper we consider nonlinear Choquard equation \begin{equation*} -\Delta u+V(x)u=(I_\alpha*F(u))f(u)\quad {\rm in}\ \mathbb{R}^{N}, \end{equation*} where $V\in C(\mathbb{R}^N)$, $I_\alpha$ denotes the Riesz potential, $f(t)=|t|^{p-2}t+|t|^{q-2}t$ for all $t\in\mathbb{R}$, $N\geqslant5$ and $\alpha\in(0,N-4)$. Under suitable conditions on $V$, we obtain that the Choquard equation with doubly critical growth nonlinearity, i.e., $p=(N+\alpha)/N,q=(N+\alpha)/(N-2)$, has a nonnegative ground state solution by variational methods