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Existence for elliptic equations in \(L^1\) having lower order terms with natural growth. (English) Zbl 0963.35068

Summary: We deal with the following type of nonlinear elliptic equations in a bounded subset \(\Omega\subset \mathbb{R}^N\): \[ \begin{cases} -\text{div}(a(x, u,\nabla u))+ g(x,u,\nabla u)= \chi\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{P} \] 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 \(\mathbb{R}\). 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 \(W^{1,q}_0(\Omega)\) for every \(q< {N\over N-1}\), by adapting to the framework of \(L^1\) data a technique used in [L. Boccardo, F. Murat and J. P. Puel, Existence de solutions non bornées pour certaines equations quasi-linéaires, Port. Math., 41, 507-534 (1982; Zbl 0544.35040)], which simply relies on the Fatou lemma combined with the sign assumption on \(g\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A10 Cauchy-Kovalevskaya theorems
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Citations:

Zbl 0544.35040
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