Existence of multiple solutions for quasi-linear degenerate elliptic equations

The present paper is concerned with a class of quasi-linear elliptic degenerate equations. The degenerate operator arises from analysis of manifolds with singularities. The variational methods are applied here to verify the existence of infinitely many solutions for the problem.

     

Finding multiple solutions to elliptic systems with polynomial nonlinearity

Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L² norm and H¹ norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D₄ symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.     

Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations.     

Qualitative properties and classification of solutions to elliptic equations with Stein-Weiss type convolution part

In this paper, we study the qualitative properties and classification of the solutions to the elliptic equations with Stein-Weiss type convolution part. Firstly, we study the qualitative properties, such as the symmetry, regularity and asymptotic behavior of the positive solutions. Secondly, we classify the non-positive solutions by proving some Liouville type theorems for the finite Morse index solutions and stable solutions to the nonlocal elliptic equations with double weights.

     

Un problema variazionale per soluzioni radiali di equazioni ellittiche estremanti

Sunto Si risolve un problema variazionale riguardante un operatore differenziale ordinario non lineare del secondo ordine, con una singolarità nell’origine. Il problema ha origine dalla ricerca di limitazioni puntuali per soluzioni di equazioni ellitiche non variazionali.

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Summary A variational problem concerning a second order nonlinear ordinary differential operator with a singularity in the origin is solved. The problem arises from the research for pointwise estimates for solutions of non variational elliptic equations.

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Lavoro eseguito nell’ambito del contratto ? Equazioni funzionali ? del Consiglio Nazionale delle Ricerche.

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Entrata in Redazione il 30 novembre 1970.     

A priori estimates for the difference of solutions to quasi-linear elliptic equations

The purpose of this paper is to prove a priori estimates for the difference of weak solutions to certain quasi-linear elliptic equations, where the p(x)-Laplace operator serves as a prototype for our method.     

Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems.     

Unloading near a shear band: a free boundary problem for the wave equation

ABSTRACT

For solutions on unbounded domains of boundary value problems for a class of quasilinear elliptic equations which are not uniformly elliptic, we prove that the solutions have the same bounds as those of the boundary data.     

Abstract weak Harnack inequality, multiple fixed points and p-Laplace equations

This paper presents an abstract theory for the existence, localization and multiplicity of fixed points in a cone. The key assumption is the property of the nonlinear operator of satisfying an inequality of Harnack type. In particular, the theory offers a completely new approach to the problem of positive solutions of quasilinear elliptic equations with p-Laplace operator.     

Riesz kernels and pseudodifferential operators attached to quadratic forms over p-adic fields

We study pseudodifferential equations and Riesz kernels attached to certain quadratic forms over p-adic fields. We attach to an elliptic quadratic form of dimension two or four a family of distributions depending on a complex parameter, the Riesz kernels, and show that these distributions form an Abelian group under convolution. This result implies the existence of fundamental solutions for certain pseudodifferential equations like in the classical case.     

The energy method for constructing time-harmonic solutions to the maxwell equations

We propose some minimum principle for an energy functional in an elliptic boundary value problem that arises in constructing time-harmonic solutions to the Maxwell equations. We suggest the potentials other than the vector and scalar potentials, used in the mathematical modeling of electromagnetic fields since the operators of traditional problems are not sign definite, which complicates constructions of iterative solution methods. We consider the problem in a parallelepiped whose boundary is ideally conducting. For nonresonant frequencies we prove that the operator of the boundary value problem is positive definite, propose a minimum principle for a quadratic energy functional, and prove the existence and uniqueness of generalized solutions.     

Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity

ABSTRACT

The existence of a nontrivial solution for quasi-linear elliptic equations involving the p-Laplace operator and a nonlinearity with p-linear growth at infinity is proved. Techniques of Morse theory are employed.     

Solvability and properties of solutions of nonlinear elliptic equations

The paper contains an exposition of variational and topological methods of investigating general nonlinear operator equations in Banach spaces. Application is given of these methods to the proof of solvability of boundary-value problems for nonlinear elliptic equations of arbitrary order, to the problem of eigenfunctions, and to bifurcation of solutions of differential equations. Results are presented of investigations of the properties of generalized solutions of quasilinear elliptic equations of higher order.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 9, pp. 131–254, 1976.     

Existence of non-radially symmetric viscosity solutions to semilinear degenerate elliptic equations with radially symmetric coefficients in the plane, Part I

We prove the existence of non-radially symmetric solutions for semilinear degenerate elliptic equations with radially symmetric coefficients in the plane. We adapt the viscosity solution for the weak solution. The key arguments consist of the analysis of the structure of 2π-periodic solutions for the associated Laplace-Beltrami operator and construction of super- and sub-solutions which have the prescribed asymptotic structures.     

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.     

Coupled-Mode Equations and Gap Solitons in a Two-Dimensional Nonlinear Elliptic Problem with a Separable Periodic Potential

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schrödinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier–Bloch decomposition and the implicit function theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a nondegeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross–Pitaevskii equation approximated by solutions of the coupled-mode equations are obtained for a finite-time interval.     

Set-valued mappings specified by regularization of the Schrödinger equation with degeneration

The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.     

A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations

This paper is concerned with an operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.     

Backward SDEs and Cauchy problem for semilinear equations in divergence form

 We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces. Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002 Research supported by KBN Grant 0253 P03 2000 19. Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55 Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution     

New Linking Theorem and Sign-Changing Solutions

Abstract

In this paper, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A general liking type theorem is given with the location of the critical point produced specified in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities, and may or may not be resonant with respect to Fu?ík spectrum.