Geodesic
Existence of solutions of certain quasilinear elliptic equations in $\mathbb R^N$ without conditions at infinity
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 133-147 
G. I. Laptev. Existence of solutions of certain quasilinear elliptic equations in $\mathbb R^N$ without conditions at infinity. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 133-147. http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a8/
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     title = {Existence of solutions of certain quasilinear elliptic equations in~$\mathbb R^N$ without conditions at infinity},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with conditions for the existence of solutions of the equations $$ -\sum_{i=1}^nD_iA_i(x,u,Du)+A_0(x,u)=f(x),\quad x\in\mathbb R^n, $$ considered in the whole space $\mathbb R^n$, $n\ge2$. The functions $A_i(x,u,\xi)$, $i=1,\dots,n$, $A_0(x,u)$, and $f(x)$ can arbitrarily grow as $|x|\to\infty$. These functions satisfy generalized conditions of the monotone operator theory in the arguments $u\in\mathbb R$ and $\xi\in\mathbb R^n$. We prove the existence theorem for a solution $u\in W_{\mathrm{loc}}^{1,p}(\mathbb R^n)$ under the condition $p>n$.

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