Existence of solutions for the Dirichlet problem of some degenerate semilinear elliptic equations (original) (raw)

On the Dirichlet problem for a class of nonlinear degenerate elliptic equations

Contributions to Mathematics, 2020

In this paper, we prove the existence and uniqueness of solutions for a class of nonlinear degenerate elliptic equations L u(x) = f0(x) − n j=1 Dj fj(x) in the setting of the weighted Sobolev spaces, where Dj = ∂/∂xj and L is a second order degenerate elliptic operator in divergence form in a bounded open subset of R n .

Existence of solutions for some degenerate quasilinear elliptic equations

Le Matematiche, 2009

In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations − n ∑ j=1 D j [ω(x)A j (x, u, ∇u)] + ω(x)g(x, u(x), ∇u(x)) = f 0 − n ∑ j=1 D j f j , on Ω, in the setting of the weighted Sobolev spaces W 1,p 0 (Ω, ω).

Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

Moroccan Journal of Pure and Applied Analysis, 2019

In this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations { - div [ 𝒜 ( x , ∇ u ) ω 1 + 𝒝 ( x , u , ∇ u ) ω 2 ] = f 0 ( x ) - ∑ j = 1 n D j f j ( x ) in Ω , u ( x ) = 0 on ∂ Ω , \left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right. in the setting of the weighted Sobolev spaces.