Nonhomogeneous Operator Research Articles

Existence and non-existence of positive solution for (p, q)-Laplacian with singular weights

We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 < q < p < \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ > 0 there exists an interval of eigenvalues for our eigenvalue problem.

Two constant sign solutions for a nonhomogeneous Neumann boundary value problem

Abstract We consider a nonlinear Neumann problem with a nonhomogeneous elliptic differential operator. With some natural conditions for its structure and some general assumptions on the growth of the reaction term we prove that the problem has two nontrivial solutions of constant sign. In the proof we use variational methods with truncation and minimization techniques.

Existence of solutions to a class of asymptotically linear Schrödinger equations in [formula omitted] via the Pohozaev manifold

In this study, we investigate the nonexistence of a least energy solution and the existence of a positive solution for a class of nonhomogeneous asymptotically linear Schrödinger equations in Rn via the Pohozaev manifold. After changing the variables, the quasilinear operator becomes a semilinear nonhomogeneous operator. The technique used employs variational methods that are constrained to the Pohozaev manifold, which are combined with the splitting lemma.

Classification of isolated singularities for nonhomogeneous operators in divergence form

Consider the equation div(φ(|∇u|)|∇u|∇u)=0 on the punctured unit ball from RN (N≥2), where φ is an odd, increasing homeomorphism from R onto R of class C1. Under reasonable assumptions on φ we prove that if u is a non-negative solution of our equation, then either 0 is a removable singularity of u or u behaves near 0 as the fundamental solution of the equation investigated here. In particular, our result complements to the case on nonhomogeneous operators in divergence form Bôcher's Theorem and some classical results by Serrin.

The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Existence and multiplicity of solutions for equations involving nonhomogeneous operators of "Equation missing" -Laplace type in "Equation missing"

Abstract We are concerned with the following elliptic equations with variable exponents: − div ( φ ( x , ∇ u ) ) + | u | p ( x ) − 2 u = λ f ( x , u ) in R N , where the function φ ( x , v ) is of type | v | p ( x ) − 2 v with continuous function p : R N → ( 1 , ∞ ) and f : R N × R → R satisfies a Carathéodory condition. The purpose of this paper is to show the existence of at least one solution, and under suitable assumptions, infinitely many solutions for the problem above by using mountain pass theorem and fountain theorem. MSC: 35D30, 35J60, 35J90, 35P30, 46E35.

Three solutions for equations involving nonhomogeneous operators of p-Laplace type in R N

In this paper, we are concerned with the following elliptic equation where the function is of type and satisfies a Carathéodory condition. We establish the existence of at least three weak solutions for the problem above which is based on an abstract three critical points theory due to Ricceri. Moreover, we determine precisely the intervals of λ’s for which the given problem possesses either only the trivial solution or at least two nontrivial solutions. MSC:35D30, 35D50, 35J15, 35J60, 35J62.

Построение трансляционных матриц для неоднородных дифференциальных операторов

The construction algorithm of transfer matrices for nonhomogeneous differentiation operators on example of Poisson equation in spherical coordinates is presented. Design formulas to estimation of unknown fields in separate slabs of laminated sphere are obtained. This method allows you to transfer the compatibility conditions with layer upon layer, eliminating unnecessary arbitrary constants without additional solutions of algebraic systems of sufficiently high order.

Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

By variational methods and Morse theory, we prove the existence of uncountably many $$(\alpha ,\beta )\in \mathbb R ^2$$ for which the equation $$-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}$$ in $$\Omega $$ , has a sign changing solution under the Neumann boundary condition, where a map $$A$$ from $$\overline{\Omega }\times \mathbb R ^N$$ to $$\mathbb R ^N$$ satisfying certain regularity conditions. As a special case, the above equation contains the $$p$$ -Laplace equation. However, the operator $$A$$ is not supposed to be $$(p-1)$$ -homogeneous in the second variable. In particular, it is shown that generally the Fucik spectrum of the operator $$-\mathrm{div}\, A(x, \nabla u)$$ on $$W^{1,p}(\Omega )$$ contains some open unbounded subset of $$\mathbb R ^2$$ .

On the solutions of the [formula omitted]-Laplacian problem at resonance

We consider a resonance problem driven by a nonhomogeneous operator and a Carathéodory reaction f(.,.). Using a variant of the monotone operator theorem, we show that the problem has at least a nontrivial solution.

Global solution branches for equations involving nonhomogeneous operators of [formula omitted]-Laplace type

It is shown that if μ is not an eigenvalue of an associated p -Laplacian, then the equation − div ( φ ( x , ∇ u ) ) = μ | u | p − 2 u + f ( λ , x , u , ∇ u ) with nonhomogeneous φ (which is assumed to behave asymptotically as the function generating the associated p -Laplacian) has a global branch of solutions ( λ , u ) . Also the case of modified p -Laplace operators and generalizations thereof are discussed.

Global bifurcation for equations involving nonhomogeneous operators in an unbounded domain

We study the structure of the set of solutions of a nonlinear equation involving nonhomogeneous operators: − div ( ϕ ( x , | ∇ u | ) ∇ u ) = μ 0 g ( x ) | u | p − 2 u + f ( λ , x , u ) in R N satisfying certain conditions on ϕ , g and f when μ 0 is not an eigenvalue of the p -Laplacian in some sense. This is based on a bifurcation result on noncompact connected sets of solutions for nonlinear operator equations.

Powers of the Parabolic Dirac Operator

The following paper deals with the characterization of the polynomial null solutions for a homogeneous and a non-homogeneous parabolic Dirac operator, via Cauchy-Riemann-like systems. For that we will use the factorizations of the Laplace and the heat operator, respectively. We will present a Fischer decomposition for the case of homogeneous polynomials. At the end of the paper two examples are presented.

Crossing of eigenvalues: Measure type estimates

This paper deals with the behavior of eigenvalues for some non-homogeneous elliptic operators. More precisely, we present measure type estimates evaluating neighborhoods of the so-called resonant set. Various problems like the non-homogeneous incompressible limit of the compressible Navier–Stokes equations lead to such studies. This will be the purpose of a forthcoming paper.

On The Second Eigenvalue For Nonhomogeneous Quasi-linear Operators

In this paper we study a nonlinear eigenvalue problem and associated perturbations of the problem. More specifically, we generalize a variational characterization of the second eigenvalue for homogenous quasi-linear elliptic operators, such as the p-Laplacian, to a class of nonhomogeneous quasi-linear elliptic operators. Neumann boundary data is assumed throughout the paper. To demonstrate the utility of this characterization we use it to prove a generalized Fredholm alternative for nonresonant perturbations of the given eigenvalue problem.

Polynomial expansions of boolean functions in images of nonhomogeneous operators

To represent the functions in terms of which expansions are constructed, the so-called dp-, pt -, and dt -operators are used. The existence of polynomial expansions of Boolean functions in functions obtained by applying heterogeneous operators of the above-mentioned types to a nondegenerate Boolean function is proved. Some methods of finding coefficients of these expansions are also considered.

Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups

The purpose of this article is to extend our investigations on the problem of solvability for second order lefl-invariant differential operators on twostep nilpotent Lie groups from the case of homogeneous operators, which was treated in a complete way in [MR3], to the case of non-homogeneous operators. More precisely, on a connected, simply connected two-step nilpotent Lie group G we shall consider lefi-invariant differential operators of the form

Solvability of Transversally Elliptic Differential Operators on Nilpotent Lie Groups

1. Introduction. In this paper, we study some aspects of solvability of left invariant differential operators on connected, simply connected nilpotent Lie groups. In Part I, we prove a general necessary condition for microlocal solvability and apply it to the case of a transversally elliptic operator on a 2-step nilpotent Lie group. We show in Part II that for a special class of 2-step groups, the criterion of Part I is also sufficient for local solvability; the main argument here involves the division of distributions by analytic functions. Finally, Part III is devoted primarily to a sufficient condition for the existence of a global fundamental solution; this condition is satisfied for the operators discussed in Part II. We now describe these results in more detail. Let G be a (connected, simply connected) 2-step nilpotent Lie group with Lie algebra g. Then g = g1 0 g2 (a vector space direct sum) with g2 = [g, g] = [g1, gl ]. This grading of g induces a grading of the universal enveloping algebra u(g), u(g) = (30=o u,(g). Thus a left invariant differential operator L E u(g) can be written L = ET0 LL, with Li homogeneous of degree j. In the homogeneous case (L = Lm), criteria for local solvability of L have been given by the authors individually ([21], [3], [4]) and severally ([5]), by Levy-Bruhl ([12], [13], [14]), by Rothschild-Tartakoff ([24]), and by others; a survey of some of these results is found in [18]. In Parts I and II of this paper, we use new techniques to obtain results for non-homogeneous operators. We say that L is tranisversally elliptic (or elliptic in the genieraitinlg directions) if for every nontrivial 1-dimensional unitary representation u of G, a(Lm) * 0. (Note that a 0 on g2.) A differential operator P is locally solvable at xo if there is a neighborhood U of x0 such that the equation

The Condition of Ordinary Integral Convergence in the Asymptotic Theory of Linear Differential Equations With Almost Constant Coefficients

In this paper, we continue the work started in [2], wherethe asymptotic integration of nth order linear differential equations was considered under smallness conditions, expressed in terms of ordinary integral convergence. The method of integration in [2] was based on using the Banach contraction principle for a linear nonhomogeneous operator in a Banach space. In this paper, we use the same principle for a nonlinear operator which corresponds to the equation for the logarithmic derivative of the original dependent variable. The result proved concerns a class of equations considered in [2] under essential additional restrictions.