The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

Abstract

We prove the existence of extremal solutions of the following quasilinear elliptic problem − ∑ =1 ( / ) ( , ( ), ( )) + ( , ( ), ( )) = 0 under Dirichlet boundary condition in Orlicz-Sobolev spaces 1 0 (Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term : Ω × R × R → R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.

Existence and regularity results for nonlinear elliptic equations in Orlicz spaces

“…Existence (and regularity) results for problems with A(∇u) = a(|∇u|)∇u can be found in [5,6,10,11,18,23,29]. In [25] the authors deal with an operator depending on the three variables via Young functions of a real variable.…”

“…We stress that Young's functions are also involved in the growth of the convective term f . Similar hypotheses can be found in [10,11,25]. Given the non-variational nature of the problem, we use the method of sub and super solutions, togheter with truncation techniques and the theorem of existence of zeros for monotone operators.…”

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

“…We refer to some results in variable exponent Sobolev or Orlicz-Sobolev spaces [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”