Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro

Displaying similar documents to “Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations”

Existence of solutions for Navier problems with degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2015)

Communications in Mathematics

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In this paper we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations ${Δ (v (x) | Δ u |}^{q - 2} Δ u) - \sum_{j = 1}^{n} D_{j} [ω (x) 𝒜_{j} (x, u, \nabla u)] = f_{0} (x) - \sum_{j = 1}^{n} D_{j} f_{j} (x), in Ω$ in the setting of the weighted Sobolev spaces.

Uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Applicationes Mathematicae

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We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $- \sum_{i, j = 1}^{n} D_{j} (a_{i j} (x) D_{i} u (x)) + b (x) u (x) + d i v (Φ (u (x))) = g (x) - \sum_{j = 1}^{n} f_{j} (x)$ on Ω in the setting of the space H₀(Ω).

$L^{p}$ -Estimates for linear elliptic systems with discontinuous coefficients

Filippo Chiarenza, Michelangelo Franciosi, Michele Frasca (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this Note we give $L^{p}$ estimates $(1 < p < + \infty)$ for the highest order derivatives of an elliptic system in non-divergence form with coefficients in VMO.

Existence and uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Archivum Mathematicum

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In this article we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations $\begin{matrix} Δ (v (x) {| Δ u |}^{p - 2} Δ u) & - \sum_{j = 1}^{n} D_{j} [ω (x) 𝒜_{j} (x, u, \nabla u)] \\ = & f_{0} (x) - \sum_{j = 1}^{n} D_{j} f_{j} (x), i n Ω \end{matrix}$ in the setting of the weighted Sobolev spaces.

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

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In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the...

Remarks on the Existence of Many Solutions of Certain Nonlinear Elliptic Equations

Edward N. Dancer, Shusen Yan (2007)

Bollettino dell'Unione Matematica Italiana

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We show how a change of variable and peak solution methods can be used to prove that a number of nonlinear elliptic partial differential equations have many solutions.

Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations

Francesca Da Lio, Boyan Sirakov (2007)

Journal of the European Mathematical Society

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We study uniformly elliptic fully nonlinear equations $F (D^{2} u, D u, u, x) = 0$ , and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.

On positive solutions of quasilinear elliptic systems

Yuanji Cheng (1997)

Czechoslovak Mathematical Journal

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In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems $\}\begin{matrix} - Δ_{p} u = f (x, u, v), & in Ω, - Δ_{p} v = g (x, u, v), & in Ω, u = v = 0, & on \partial Ω, \end{matrix}$ where $- Δ_{p}$ is the $p$ -Laplace operator, $p > 1$ and $Ω$ is a $C^{1, α}$ -domain in $ℝ^{n}$ . We prove an analogue of [7, 16] for the eigenvalue problem with $f (x, u, v) = λ_{1} v^{p - 1}$ , $g (x, u, v) = λ_{2} u^{p - 1}$ and obtain a non-existence result of positive solutions for the general systems.

Integral inequalities and summability of solutions of some differential problems

Lucio Boccardo (2000)

Banach Center Publications

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The aim of this note is to indicate how inequalities concerning the integral of ${| \nabla u |}^{2}$ on the subsets where |u(x)| is greater than k ( $k \in I R^{+}$ ) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of ${| \nabla u |}^{2}$ on the subsets where |u(x)| is less...

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $- \sum_{i, j = 1}^{n} D_{i} (a_{i j} D_{j} u) = f (u, \int_{Ω} g (u^{p}))$ , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps into spheres.

Bendahmane, Mostafa, Karlsen, Kenneth H. (2006)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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