Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces

Abstract

UDC 517.5

We deal with the existence result for nonlinear elliptic equations related to the form
$$
Au + g(x, u,\nabla u) = f,
$$
where the term $-{\rm div}\Big(a(x,u,\nabla u)\Big)$ is a Leray–Lions operator from a subset of $W^{1}_{0}L_M(\Omega)$ into its dual.  The growth and coercivity conditions on the monotone vector field $a$ are prescribed by an $N$-function $M$ which does not have to satisfy a $\Delta_2$-condition.
Therefore we use Orlicz–Sobolev spaces which are not necessarily reflexive and assume that the nonlinearity $g(x,u,\nabla u)$ is a Carathéodory function satisfying only a growth condition with no sign condition.
The right-hand side~$f$ belongs to $W^{-1}E_{\overline{M}}(\Omega).$

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Published
28.03.2020
How to Cite
Moussa H., RhoudafM., and Sabiki H. “Existence Results for a Perturbed Dirichlet Problem Without Sign Condition in Orlicz Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 4, Mar. 2020, pp. 509-26, doi:10.37863/umzh.v72i4.373.
Section
Research articles