Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

<abstract><p>In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $ \lambda $ and $ \mu $ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $ \lambda $, and analyze the relationship between $ \lambda $ and $ \mu $. In the end, we investigate the concentration phenomenon of solutions when $ \mu\to 0 $, and obtain a specific lower bound of the parameter $ \lambda $ which is independent of $ \mu $.</p></abstract>

Finite Element Analysis of Contact Stress for Crowned Cylindrical Roller Bearings of Railway Passenger Cars

The finite element model of NJP3226X1 cylindrical roller bearings was established by ABAQUS software, and the contact stress distribution of three kinds of modification curve(linear, circular-arc, logarithmic) were compared. The results showed that the maximal contact stress of logarithmic modified roller is the minimum, its stress distribute uniformly, and obvious stress concentration phenomenon does not exist, next is circular-arc modified roller, which stress at the two ends is too small and the stress in the middle is large, linear roller has the highest contact stress, and there is an obvious stresses concentration phenomenon at both ends. Through the analysis of the contact stress distribution of the logarithmic shaped roller under different repair quantities, the reasonable repair quantities are obtained.

Eigenvalues of the Laplacian Operator and Neighborhood Enlargement for Compact Manifolds

In this investigation and under the conception of measure concentration phenomenon we found that the enlargement of the neighborhood for an n – dimensional compact Riemannian manifolds (M,g) relative to the eigenvalues λ of the Laplace operator ∆ on (M,g). And we found that r~1/√λ. تناول هذا البحث تجسيم الجوار لمتعدد الطيات المتراص في البعد n وذلك باستخدام مفهوم تركيز الحجم. كما وجدنا أن نصف القطر r لتجسيم الجوار لمتعدد الطيات يرتبط مع القيم الذاتية λ لمؤثر لابلاس Δ على متعدد الطيات. الكلمات المفتاحية: نصف قطر تجسيم الجوار، متباينات متساوي المقاييس، تركيز الحجم، مؤثر لابلاس، القيم الذاتية لمؤثر لابلاس.