Existence for Elliptic Equations in $L^{1}$ having Lower Order Terms with Natural Growth

Alessio Porretta

Abstract: We deal with the following type of nonlinear elliptic equations in a bounded subset $\Omega\subset\R^N$:
$$ \cases{-\div(a(x,u,\nabla u)))+g(x,u,\nabla u)=\chi &in $\Omega$,\cr
u=0 &on $\partial\Omega$,\cr} \leqno{({\rm P})} $$
where both $a(x,s,\xi)$ and $g(x,s,\xi)$ are Carathéodory functions such that $a(x,s,\cdot)$ is coercive, monotone and has a linear growth, while $g(x,s,\xi)$ has a quadratic growth with respect to $\xi$ and satisfies a sign condition on $s$, that is $g(x,s,\xi)s\geq 0$ for every $s$ in $\R$. The datum $\chi$ is assumed in $L^1(\Omega)+H^{-1}(\Omega)$. We prove the existence of a weak solution $u$ of (P) which belongs to the Sobolev space $W_0^{1,q}(\Omega)$ for every $q<\frac N{N-1}$, by adapting to the framework of $L^1$ data a technique used in [6], which simply relies on Fatou lemma combined with the sign assumption on $g$.

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