Concave-convex Type Research Articles

Multiplicity of nonnegative solutions to a weighted indefinite elliptic equation of the concave–convex type

Multiplicity of nonnegative solutions to a weighted indefinite elliptic equation of the concave–convex type

Double phase anisotropic variational problems involving critical growth

Abstract In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the p ( ⋅ ) p\left(\cdot ) -Laplace equations.

W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) versus C 0 1 ( Ω ) × C 0 1 ( Ω ) local minimizers

In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H ( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω, for some C > 0, ∀ t , s ∈ R, p < σ i ⩽ p ∗ : = N p / ( N − p ), i = 1 , 2, and 1 < p < N. We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ). An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.

Elliptic equations with critical and supercritical growth at the boundary

In this study, we consider two classes of elliptic problems with nonlinear boundary conditions of concave–convex type. In the first problem, we obtain two nonzero and nonnegative solutions when the nonlinear term exhibits critical growth. In the second problem, we obtain infinitely many solutions (with no prescribed sign) by assuming that the nonlinearity is even and subcritical near the origin but with no growth condition at infinity.

On eigenvalues of p-biharmonic operator and associated concave-convex type equation

In this article, we are interested in the simplicity and the existence of the first eigensurface for the third order spectrum of p-biharmonic operator plus potential with weights and the existence of multiple solutions for associated concave-convex type equation under Navier boundary conditions.

Schrödinger p⋅–Laplace equations in RN involving indefinite weights and critical growth

We study a class of critical Schrödinger p⋅–Laplace equations in RN, with reaction terms of the concave–convex type and involving indefinite weights. The class of potentials used in this study is different from that in most existing studies on Schrödinger equations in RN. We establish a concentration-compactness principle for weighted Sobolev spaces with variable exponents involving the potentials. By employing this concentration-compactness principle and the Nehari manifold method, we obtain existence and multiplicity results for the solution to our problem.

Cascade Failure Model in Multimodal Transport Network Risk Propagation

The cascade failure theory is introduced into the risk propagation problem of the multimodal transport network in order to study the inherent law of risk propagation and provide theoretical support for the safety management of multimodal transport networks. Firstly, this paper analyses the characteristics of the multimodal transport network and concludes that the risk of the multimodal transport network belongs to failure risk. Secondly, the applicability of cascade failure theory is expounded. Based on cascade failure theory, a risk propagation model of the multimodal transport network is established. Through simulation experiments, the risk propagation of the multimodal transport network is analyzed from the differences of node distribution and node type. The process is analyzed, and the results show that different node distributions and different types of risk source nodes will have an impact on the risk propagation process. The influence of four types of node distributions on the risk propagation effect is in the following order: increasing type > concave-convex type ≈ balanced type > decreasing type. The influence of four types of source nodes on the risk propagation effect is in the following order: transportation type > transporting type > storage type > assistant type.

Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

Abstract We establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

Energy-minimum optimization of the intelligent excavating process for large cable shovel through trajectory planning

Large cable shovel (LCS) is a complex engineering machine which is widely used in the open pit mine. It is characterized by low efficiency, high maintenance cost, and high energy consumption if manipulated by an inexperienced operator. To address these challenges, intelligentization could be a feasible solution. In this work, an intelligent excavation system is put forward and the corresponding energy-minimum optimization through trajectory planning of the optimal excavation is developed to realize the intelligentization of the LCS. Firstly, the excavating resistance acting on the dipper is modeled and the corresponding forces are analyzed. Then, by establishing the kinetics and dynamic models of the excavating process, the point to point (PTP) trajectory planning method is developed by setting the objective to minimize the energy consumption per unite volume material. Polynomial curves in different degrees are used in the PTP planning method and the optimal one is compared with the conventional S-curve in terms of the excavating performance. To explore the advantage of the proposed intelligent system and the corresponding trajectory planning based energy-minimum optimization method, four types of ore piles with different pile angles are compared with respect to the excavating performance. Results show that the larger the pile angle is, the later the maximum hoist power and crowd power will appear. Further, the effects on the excavating performance from different ore piles with complex terrains, including the flat type, concave type, convex type, and concave-convex type, are also studied in the numerical experiments. It is found that the trajectory based energy-minimum optimization method for the intelligent LCS can significantly save excavation energy as well as keep sufficient fill factor.

Positivity results for indefinite sublinear elliptic problems via a continuity argument

We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some positivity results for indefinite concave-convex type problems.

Multiplicity of solutions of some quasilinear equations in ${\mathbb{R}^{N}}$ with variable exponents and concave-convex nonlinearities

We prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.

Comparative Analysis of Curvilinear Gears With Produced by the New Production Method

Abstract In this study, we describe the finite element model analysis of novel pinion gears having a curvilinear teeth profile which were previously manufactured by us for the first time with the highest accuracy. Herringbone and spur gears were also modeled by using the same approach in order to compare the advantages of each, both under actual-working- and stress-conditions. Additionally, these three gear types were compared through noise analysis for the first time as well. All these analyses suggest that the concave–convex type gears display superiority in load resistance and show reasonably low levels of noise.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR DEGENERATE $p(x)$-LAPLACE EQUATIONS INVOLVING CONCAVE-CONVEX TYPE NONLINEARITIES WITH TWO PARAMETERS

We show the existence of two nontrivial nonnegative solutions and infinitely many solutions for degenerate $p(x)$-Laplace equations involving concave-convex type nonlinearities with two parameters. By investigating the order of concave and convex terms and using a variational method, we determine the existence according to the range of each parameter. Some Caffarelli-Kohn-Nirenberg type problems with variable exponents are also discussed.

Multiplicity of solutions to a nonlinear boundary value problem of concave–convex type

Problem (P){−Δpu+|u|p−2u=|u|r−1ux∈Ω|∇u|p−2∂u∂ν=λ|u|s−1ux∈∂Ω, where Ω⊂RN is a bounded smooth domain, ν is the unit outward normal at ∂Ω, Δp is the p-Laplacian operator and λ>0 is a parameter, was studied in Sabina de Lis (2011) and Sabina de Lis and Segura de León (in press). Among other features, it was shown there that when exponents lie in the regime 1<s<p<r, a minimal positive solution exists if 0<λ≤Λ, for a certain finite Λ, while no positive solutions exist in the complementary range λ>Λ. Furthermore, in the radially symmetric case a second positive solution exists for λ varying in the same full range (0,Λ) provided r<p∗. Our main achievement in this work just asserts that such global multiplicity feature holds true when Ω is a general domain. To show such result the well-known Brezis–Nirenberg variational result in Brezis and Nirenberg (1993) must be extended to the framework of (P). This is the second main contribution in the present work.

Research on the Simulation of Textile Fabric Pattern Designs Based on Digital Image Processing Technology

With the diversification tendency of clothing art, this paper mainly researches on the simulation of concave-convex type carvings on different fabrics by employing some cartoon characters such as Hello Kitty as direct patterns. Spectrum reconstruction processes are carried out on both the vivid characters and the fabrics with different materials and textures so as to integrate the favorable features of both sides. This paper also combines the image processing technology with the computer simulation technology, and sets cartoon elements as one of the pop cultures’ label. Its application in clothing keeps in line with the modern fashion trends, therefore creating a profound influence in this field. Moreover, the application of cartoon characters to the simulation research of textile fabric pattern designs is highly innovative, vivid, and has a very broad potential market.

Existence and nonexistence of positive solutions for parametric Neumann problems with $p$-Laplacian

Using variational methods based on the critical point theory and suitable truncation and comparison techniques, we study existence, multiplicity and nonexistence of positive solutions for a parametric nonlinear Neumann problem driven by the $p$-Laplacian. Our hypotheses cover the case of nonlinearities of concave-convex type whose exponents depend on the parameter.

On some indefinite and non-powerlike elliptic equations

We consider a class of elliptic problems for the operator −Δ+V, acting on H01(Ω), where Ω is a bounded domain. The main feature here is the indefinite behavior of both the operator and the nonlinearity. Based on the global geometry of the energy functional, we prove the existence of two local minimizers and a mountain-pass critical point for a concave–convex type nonlinearity. Subcritical and critical growth are allowed on the nonlinearity.

Lack of coercivity in a concave–convex type equation

Existence and multiplicity of non-negative solutions are investigated for the concave–convex type equation $$-\Delta_p u +V(x)u^{p-1}=\lambda a(x) u^{r-1}+b(x)u^{q-1},\quad u\in W_0^{1,p}(\Omega),$$ where Ω is a bounded domain and 1 < r < p < q < p*. By minimization on the Nehari manifold, we find conditions on V, a, and b that yield up to four non-negative solutions when the left-hand side of the equation has a non-coercive behavior, a and b are sign-changing, and λ is positive and sufficiently small.

On a fourth-order quasilinear elliptic equation of concave–convex type

We consider a fourth-order quasilinear equation depending on a nonnegative parameter λ and with subcritical or critical growth. Such equation is equivalent to a Hamiltonian system and the main goal of this work is to prove the existence of at least two positive and infinitely many solutions for such equation when the parameter λ is positive and small enough.

Multiplicity Results forp-Laplacian with Critical Nonlinearity of Concave-Convex Type and Sign-Changing Weight Functions

The multiple results of positive solutions for the following quasilinear elliptic equation: in on , are established. Here, is a bounded smooth domain in denotes the -Laplacian operator, is a positive real parameter, and are continuous functions on which are somewhere positive but which may change sign on . The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.