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Nonlinear elliptic equations in $\bbfR\sp N$ without growth restrictions on the data. (English) Zbl 0810.35021
The authors examine equations of the form (*) $-\text{div } A(x,Du)+ g(x,u) =f$, where the principal part of the operator, $-\text{div } A(x,Du)$ acts like the $p$-Laplacian. Provided $ug(x,u)$ goes to infinity fast enough as $\vert u\vert\to\infty$, they show that equation (*) has a distributional solution in $\bbfR\sp N$ for any locally integrable function $f$. In addition, the gradient of this solution, which is only assumed to be locally in $L\sp{p-1}$, is locally in $L\sp q$ for some $q>p-1$; the exponent $q$ is given explicitly in terms of $p$, $N$, and the growth of $g$.

35J60Nonlinear elliptic equations
35D10Regularity of generalized solutions of PDE (MSC2000)
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