<p>In this paper, we study the existence of renormalized solutions for the singular non-coercive quasilinear elliptic problem <Equation ID="Equa"> <EquationSource Format="TEX">\(\begin{aligned} {\left\{ \begin{array}{ll} -\displaystyle \operatorname {div}a(x, u, \nabla u) =\frac{f}{u^{\theta }}-\operatorname {div}\left( \phi (u)\right) &amp; \text{ in } \Omega , \\ u=0 &amp; \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}\)</EquationSource> </Equation>in the Musielak–Orlicz Sobolev spaces, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-\operatorname {div} \left( a(x, u, \nabla u)\right)\)</EquationSource> </InlineEquation> is a degenerate Leray-Lions operator, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\phi \in C^{0}(\mathbb {R},\mathbb {R^N})\)</EquationSource> </InlineEquation>, and <i>f</i> is a nonnegative function that belongs to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^1(\Omega )\)</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(0\le \theta \le 1\)</EquationSource> </InlineEquation>. Moreover, we will conclude with some regularity results.</p>

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Renormalized solutions for some singular non-coercive quasilinear elliptic problems in Musielak–Orlicz spaces

  • Hassane Hjiaj,
  • Mohamed Sasy

摘要

In this paper, we study the existence of renormalized solutions for the singular non-coercive quasilinear elliptic problem \(\begin{aligned} {\left\{ \begin{array}{ll} -\displaystyle \operatorname {div}a(x, u, \nabla u) =\frac{f}{u^{\theta }}-\operatorname {div}\left( \phi (u)\right) & \text{ in } \Omega , \\ u=0 & \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}\) in the Musielak–Orlicz Sobolev spaces, where \(-\operatorname {div} \left( a(x, u, \nabla u)\right)\) is a degenerate Leray-Lions operator, \(\phi \in C^{0}(\mathbb {R},\mathbb {R^N})\) , and f is a nonnegative function that belongs to \(L^1(\Omega )\) , with \(0\le \theta \le 1\) . Moreover, we will conclude with some regularity results.