On the oscillation of second order quasilinear elliptic equations

By using the Riccati technique, some oscillation criteria for the second order quasilinear elliptic equation $\sum _{i,j=1}^N{D}_i\left[\Phi \left(y\right)A_{ij}\left(x\right){\left|Dy\right|}^{p-2}{D}_{j}y\right]+p\left(x\right)f\left(y\right)=0$ are established. These results contain the known oscillation theorems from the literature as special cases.     

On behavior of solutions of the quasilinear second-order parabolic equations in unbounded noncylindrical domains

The theorems of uniqueness of solutions are formulated in the classes of increasing functions for a mixed initial boundary value problem for the second-order degenerate quasiparabolic equations in unbounded noncylindrical domains. We presenta priori estimates of a special kind, analogous to the Saint-Venant principle. The proofs are based on the method of introducing a parameter.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 4, pp. 492–499, April, 1993.     

Existence of positive solutions of quasilinear degenerate elliptic equations on unbounded domains

Sunto Viene provato un teorema di esistenza di soluzioni positive per una certa classe di equazioni quasilineari ellittiche degeneri su aperti non limitati di Rⁿ utilizzando un metodo di confronta all'infinito.     

Positive unbounded solutions of second order quasilinear ordinary differential equations and their application to elliptic problems

In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.     

Existence of unbounded positive solutions of quasilinear elliptic equations in two-dimensional exterior domains

In this paper, by using fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result on unbounded positive solutions of certain quasilinear elliptic equations in two-dimensional exterior domains.     

Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.     

On the asymptotic behavior of solutions of quasilinear elliptic equations

The asymptotic behavior of solutions of second-order quasilinear elliptic and nonhyperbolic partial differential equations defined on unbounded domains inR ⁿ contained in\(\{ x_1 ,...,x_n :\left| {x_n } \right|< \lambda \sqrt {x_1^2 + ...x_{n - 1}^2 } \) for certain sublinear functions λ is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data has appropriate asymptotic behavior at infinity. We prove Phragmèn-Lindelöf theorems for large classes of nonhyperbolic operators, without «lower order terms”, including uniformly elliptic operators and operators with well-definedgenre, using special barrier functions which are constructed by considering an operator associated to our original operator. We also estimate the rate at which a solution converges to its limiting function at infinity in terms of properties of the top order coefficienta _nn of the operator and the rate at which the boundary values converge to their limiting function; these results are proven using appropriate barrier functions which depend on the behavior of the coefficients of the operator and the rate of convergence of boundary values.     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in unbounded domains

In this paper, we formulate a concentration-compactness principle at infinity which extends a result introduced by J. Chabrowski [Calc. Var. Partial Differential Equations 3 (1995) 493-512]. Then we consider some quasilinear elliptic equations in some classes of unbounded domains by solving their corresponding constrained minimization problems under certain conditions. We show the existence of positive solutions of those equations via the concentration-compactness principle at infinity, which extends some results in [Differential Integral Equations 6 (1993) 1281-1298].     

On the blow-up of positive solutions to elliptic equations in plane unbounded domains

Mathematical Notes - Using model equations of the form Δu + u σ = 0 as an example, in both linear (σ = 1) and nonlinear (σ &gt; 1) cases, we present some direct methods of...     

Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: Two approaches

In this paper, we study the asymptotic behavior as x₁→+∞ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at x₁=0 is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space RN. The second one is based on the theory of dynamical systems.     

The Dirichlet problem for second order elliptic equations with coefficients in the Stummel class in unbounded domains

In this note we prowe existence and unicity of solution of a Dirichlet problem for second order elliptic operator in the divergence form, with the coefficients of the lower order terms belonging to a variant of the Stummel-Kato class, in an unbounded domain, extending the works [6] and [2].

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Sunto In questa nota proviamo un Teorema di esistenza e unicità per la soluzione di un problema di Dirichlet relativo ad un operatore ellittico del secondo ordine in forma di divergenza, con i coefficienti dei termini di ordine inferiore appartenenti ad una variante dello spazio di Stummel-Kato, in un dominio non limitato, estendendo i lavori [6] e [2].

     

Fictitious region method for quasilinear elliptic equations of second order

Rate of convergence bounds are established for a variant of the fictitious region method for a quasilinear elliptic equation of second order with a boundary condition of the third kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 39–42, 1985.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

The oblique derivative problem for nonlinear elliptic complex equations of second order in multiply connected unbounded domains

In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem(0.1) and(0.2).