EXISTENCE AND LOCATION OF SOLUTIONS FOR A GENERAL ELLIPTIC SYSTEM WITH INTRINSIC OPERATORS

In this paper, we consider the following quasilinear Choquard equation − ϵ 2 Δ u + V ( x ) u − ϵ 2 Δ ( u 2 ) u = ϵ μ − 2 ( 1 | x | μ ∗ F ( u ) ) f ( u ) in R 2 , where ϵ > 0 is a parameter, 0 < μ < 2 , ∗ is the convolution product in R 2 , V ( x ) is a continuous real function in R 2 , F ( u ) is the primitive function of f ( u ) and f has critical exponential growth with respect to the Trudinger-Moser inequality. By employing a change of variables, the quasilinear equation can be reduced to a semilinear equation, whose associated functional is well defined in a nonstandard Orlicz space and exhibits a mountain pass geometry. Under suitable assumptions on V and f, we investigate the existence and concentration behavior of positive ground state solutions for the above problem by variational methods.

In this paper, we study the following singularly perturbed biharmonic Choquard equation: where is a parameter, , ∗ is the convolution product in , and is a continuous real function. is the primitive function of , and has critical exponential growth in the sense of the Adams inequality. By using variational methods, we establish the existence of ground state solutions when small enough.

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system [Formula: see text] where [Formula: see text], W and V are positive potentials of [Formula: see text] class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.

In this paper, we study the existence and nonexistence of solutions for some systems of semilinear elliptic equations in an unbounded domain $\Omega$. The main objective of this paper is to discuss the relationship between the existence of solutions and the translation invariance of the domain $\Omega$. More precisely, we point out how the translation invariance of our domain plays a crucial role for the existence of our problem.

In Chap. 6, on superlinear systems of Hammerstein integral equations and applications, we use the Leray–Schauder degree to obtain new results on the existence of solutions, and apply them to two-point boundary problems of systems of equations. We also are concerned with the existence of (component-wise) positive solutions for a semilinear elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing a cone K 1×K 2, which is the Cartesian product of two cones in the space \(C(\overline{\Omega})\), and computing the fixed point index in K 1×K 2, we establish the existence of positive solutions for the system.KeywordsFixed Point IndexSemilinear Elliptic SystemsHammerstein Integral EquationsLeray-Schauder DegreeSublinearThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the problem of parametric material and simultaneous topology optimization of an elastic continuum. To ensure existence of solutions to the proposed optimization problem and to enable the imposition of a deliberate maximal material grading, two approaches are adopted and combined. The first imposes pointwise bounds on design variable gradients, whilst the second applies a filtering technique based on a convolution product. For the topology optimization, the parametrized material is multiplied with a penalized continuous density variable. We suggest a finite element discretization of the problem and provide a proof of convergence for the finite element solutions to solutions of the continuous problem. The convergence proof also implies the absence of checkerboards. The concepts are demonstrated by means of numerical examples using a number of different material parametrizations and comparing the results to global lower bounds.

This thesis concerns the study of existence, nonexistence and a priori estimates of nonnegative solutions of some types of degenerate coercive and non coercive elliptic problems involving an additional term which depends on the gradient. Among other things, we obtain generalized integral conditions of Keller-Osserman type for the existence and nonexistence of solutions. Also, we show that different conditions are needed when p 2 or p 2, due to the degeneracy of the operator. The uniform a priori estimates are obtained for supersolutions and solutions of superlinear elliptic PDE or systems of such PDE in divergence form that can contain different operators and nonlinearities. We also give full boundary extensions to some "half Harnack" inequalities and quantitative Hopf lemmas, for degenerate elliptic operators like the p-Laplacian.

&lt;p style='text-indent:20px;'&gt;The present paper develops an approximation approach for solving a quasilinear Dirichlet boundary value problem that exhibits a degenerated &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian and full dependence on the solution and its gradient (convection term). The results establish that the solution set is nonempty and bounded. The principal part of the equation is driven by a &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian type operator containing a weight depending on the solution without any monotonicity assumption. The existence of a weak solution results through a limit process by means of approximate solutions arising from finite-dimensional problems. The a priori estimates are obtained in adequate Sobolev spaces with weights. An example provides an explicit illustration of the involved technique.&lt;/p&gt;

This research stems from a control problem for a suspension device. For a general class of switching stochastic mechanical systems (including closed-loop control ones), we establish the following: (1) existence and uniqueness of a weak solution and its strong Markov property, (2) mixing property in the form of the local Markov–Dobrushin condition, and (3) exponentially fast convergence to the unique stationary distribution. These results are proved for discontinuous coefficients under nondegenerate disturbances in the force field; for (3) a stability condition is additionally imposed. Linear growth of coefficients is allowed.KeywordsStochastic Differential EquationQuadratic Lyapunov FunctionStrong Markov PropertySwitching MomentSuspension DeviceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Aims/ Objectives: In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the n−dimensional elliptic system &#x0D; systems have been widely studied, but there is relatively little research on n-dimensional elliptic systems. We are very interested in this subject and want to study it. We give new conclusions on the existence, nonexistence and multiplicity of positive solutions for the n-dimensional elliptic system. &#x0D; Study Design: Study on the existence, nonexistence and multiplicity of positive solutions. Place and Duration of Study: School of Applied Science, Beijing Information Science &amp; Technology University, September 2019 to present. Methodology: We prove the existence, nonexistence and multiplicity of positive solutions by the results of fixed point index. Results: We give new conclusions of existence, nonexistence and multiplicity of positive solutionsfor the system. Conclusion: We prove the existence, nonexistence and multiplicity of positive solutions to the n-dimensional elliptic system &#x0D; and give new conclusions.

In this article, we study the existence of positive solutions for the quasilinear elliptic system −∆_p u(x) = f_1(x, v(x)) + h_1(x) in Ω,−∆_p v(x) = f_2(x, u(x)) + h_2(x) in Ω,u = v = 0 on ∂Ω,where f_i(x, s), (i = 1, 2) locates between the first and the second eigenvalues of the p-Laplacian. To prove the existence of solutions, we use a topological method the Leray-Schauder degree.

Existence and multiplicity of solutions are established, via the Variational Method, for a class of resonant semilinear elliptic system in under a local nonquadraticity condition at infinity. The main goal is to consider systems with coupling where one of the potentials does not satisfy any coercivity condition. The existence of solution is proved under a critical growth condition on the nonlinearity.

We study the existence of vector-valued solutions to quasilinear elliptic systems with nonpolynomial growth, namely N-functions. Existence of solutions is shown by means of the theory of Young measures, where the right-hand side is in the divergence form.

Solitary waves for some nonlinear Schrödinger systems