Semi-linear Second-order Elliptic Equations in $l^{1}$

$\beta(x, u)$ is not monotone but has the same sign as $u$ , we prove the existence of solutions when $f(x)$ belongs to an Orlicz class arbitrarily close to $L^{1}(\Omega)$ . We also consider equation (1) with a nonlinear boundary condition. The linear case is considered by Stampacchia [18]. Our basic technique is to multiply the equation by various monotone functions of $u$ . This method was used by Moser [14] in his proof of the DeGiorgi-Nash regularity theorem. The standard variational approach $[12, 17]$ cannot be applied to our problem for two reasons. Firstly, $\beta(x, u)$ may be rapidly increasing in $u$ and may even have vertical asymptotes. We can handle rapidly increasing nonmonotone $\beta(x, u)$ by a lemma from [20]. We can handle (multi-valued) monotone graphs by techniques from [6]. Secondly, the merely integrable function $f(x)$ need not belong to the dual space of the space where an energy estimate holds. While this work was in progress, we learned of four other related works. (i) Browder [4] allows rapidly increasing non-linear lower-order terms of high-order elliptic operators. Because his approach is variational, those of his $f’ s$ which are functions must belong to a smaller space than $L^{1}(\Omega)$ . (ii) Da Prato [7] considers equation (1) with $L=-\Delta,$ $\beta(u)$ a monotone continuous function, and $f\in L^{p}(\Omega)$ for $p>1$ . (iii) Konishi [10] has a result similar to part of our Theorem 1 in the case when $\beta(u)$ is a monotone continuous function and $\Omega$ is bounded. (His “ sub-Markov “ assumption is equivalent to our assumption (II).) His methods are entirely different from ours. (iv) Crandall’s Theorem 4.12 in [5] is closely related to our Theorem 1 in case $ L=-\Delta$ . In