Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type

Stanisław Brzychczy

Displaying similar documents to “Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type”

Existence of Solution for Quasilinear Degenerated Elliptic Unilateral Problems

Youssef Akdim, Elhoussine Azroul, Abdelmoujib Benkirane (2003)

Annales mathématiques Blaise Pascal

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An existence theorem is proved, for a quasilinear degenerated elliptic inequality involving nonlinear operators of the form $A u + g (x, u, \nabla u)$ , where $A$ is a Leray-Lions operator from $W_{0}^{1, p} (Ω, w)$ into its dual, while $g (x, s, ξ)$ is a nonlinear term which has a growth condition with respect to $ξ$ and no growth with respect to $s$ , but it satisfies a sign condition on $s$ , the second term belongs to $W^{- 1, p^{'}} (Ω, w^{*})$ .

Existence of solutions of degenerated unilateral problems with $L^{1}$ data

Lahsen Aharouch, Youssef Akdim (2004)

Annales mathématiques Blaise Pascal

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In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type $A u + g (x, u, \nabla u) = f - div F,$ where $A$ is a Leray-Lions operator and $g$ is a Carathéodory function having natural growth with respect to $| \nabla u |$ and satisfying the sign condition. The second term is such that, $f \in L^{1} (Ω)$ and $F \in Π_{i = 1}^{N} L^{p^{'}} (Ω, w_{i}^{1 - p^{'}})$ .

On a nonlocal problem for a confined plasma in a Tokamak

Weilin Zou, Fengquan Li, Boqiang Lv (2013)

Applications of Mathematics

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The paper deals with a nonlocal problem related to the equilibrium of a confined plasma in a Tokamak machine. This problem involves terms $u_{*}^{'} (| u > u (x) |)$ and $| u > u (x) |$ , which are neither local, nor continuous, nor monotone. By using the Galerkin approximate method and establishing some properties of the decreasing rearrangement, we prove the existence of solutions to such problem.

A symmetrization result for nonlinear elliptic equations.

Vincenzo Ferone, Basilio Messano (2004)

Revista Matemática Complutense

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We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x,u) + f, where the principal term is a Leray-Lions operator defined on W (Ω). The function g(x,u) satisfies suitable growth assumptions, but no sign hypothesis on it is assumed. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.

On the solvability of nonlinear elliptic equations in Sobolev spaces

Piotr Fijałkowski (1992)

Annales Polonici Mathematici

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We consider the existence of solutions of the system (*) $P (D) u^{l} = F (x, (\partial^{α} u))$ , l = 1,...,k, $x \in ℝ^{n}$ $(u = (u ¹, . . ., u^{k}))$ in Sobolev spaces, where P is a positive elliptic polynomial and F is nonlinear.

Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations

Leszek Gęba, Tadeusz Pruszko (1991)

Annales Polonici Mathematici

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This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in $Ω \subset ℝ^{n}$ , $u = D u = . . . = D^{m - 1} u$ on ∂Ω in the Sobolev space $W_{0}^{m, 2} (Ω)$ , where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.

On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions

Joachim Naumann, Jörg Wolf, Michael Wolff (1998)

Commentationes Mathematicae Universitatis Carolinae

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We prove the interior Hölder continuity of weak solutions to parabolic systems $\frac{\partial u^{j}}{\partial t} - D_{α} a_{j}^{α} (x, t, u, \nabla u) = 0 in Q (j = 1, ..., N)$ ( $Q = Ω \times (0, T), Ω \subset ℝ^{2}$ ), where the coefficients $a_{j}^{α} (x, t, u, ξ)$ are measurable in $x$ , Hölder continuous in $t$ and Lipschitz continuous in $u$ and $ξ$ .

Positive solutions of nonlinear elliptic systems

Robert Dalmasso (1993)

Annales Polonici Mathematici

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We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, $L^{\infty}$ a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.