Solvability of strongly nonlinear elliptic variational problems in weighted Orlicz–Sobolev spaces

SeMA Journal ◽  
2019 ◽  
Vol 77 (2) ◽  
pp. 119-142 ◽  
Author(s):  
Abdelmoujib Benkirane ◽  
Mostafa El Moumni ◽  
Khaled Kouhaila
Keyword(s):  
Sobolev Spaces ◽  
Variational Problems ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic

scholarly journals Existence of renormalized solutions for some quasilinear elliptic Neumann problems

2021 ◽  
Vol 8 (1) ◽  
pp. 180-206
Author(s):  
Mohamed Badr Benboubker ◽  
Hassane Hjiaj ◽  
Idrissa Ibrango ◽  
Stanislas Ouaro
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Growth Conditions ◽  
Renormalized Solutions ◽  
Neumann Problems ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Carathéodory Functions ◽  
Variable Exponent Sobolev Spaces ◽  
Quasilinear Elliptic

Abstract This paper is devoted to study some nonlinear elliptic Neumann equations of the type { A u + g ( x , u , ∇ u ) + | u | q ( ⋅ ) - 2 u = f ( x , u , ∇ u ) in Ω , ∑ i = 1 N a i ( x , u , ∇ u ) ⋅ n i = 0 on ∂ Ω , \left\{ {\matrix{ {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill \cr } } \right. in the anisotropic variable exponent Sobolev spaces, where A is a Leray-Lions operator and g(x, u, ∇u), f (x, u, ∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.

Positive solutions of strongly nonlinear elliptic problems

2015 ◽  
Vol 93 (1-2) ◽  
pp. 1-20 ◽  
Author(s):  
Francisco Julio S.A. Corrêa ◽  
Marcos L. Carvalho ◽  
J.V.A. Goncalves ◽  
Kaye O. Silva
Keyword(s):  
Positive Solutions ◽  
Elliptic Problems ◽  
Nonlinear Elliptic Problems ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic

Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents

2021 ◽  
Vol 110 (5-6) ◽  
pp. 830-841
Author(s):  
M. Mosa Al-Shomrani ◽  
M. Ben Mohamed Salah ◽  
A. Ghanmi ◽  
K. Kefi
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Variable Exponents ◽  
Nonlinear Elliptic

Diagonal non-semicontinuous variational problems

2018 ◽  
Vol 24 (4) ◽  
pp. 1333-1343
Author(s):  
Sandro Zagatti
Keyword(s):  
Sobolev Spaces ◽  
Hessian Matrix ◽  
Variational Problems ◽  
Convex Envelope ◽  
Minimum Problem ◽  
Unique Minimizer ◽  
Special Case ◽  
Minimum Points

We study the minimum problem for non sequentially weakly lower semicontinuos functionals of the form F(u)=∫If(x,u(x),u′(x))dx, defined on Sobolev spaces, where the integrand f:I×ℝm×ℝm→ℝ is assumed to be non convex in the last variable. Denoting by f̅ the lower convex envelope of f with respect to the last variable, we prove the existence of minimum points of F assuming that the application p↦f̅(⋅,p,⋅) is separately monotone with respect to each component pi of the vector p and that the Hessian matrix of the application ξ↦f̅(⋅,⋅,ξ) is diagonal. In the special case of functionals of sum type represented by integrands of the form f(x, p, ξ) = g(x, ξ) + h(x, p), we assume that the separate monotonicity of the map p↦h(⋅, p) holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.

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