## Resolution of a semilinear equation in $$L^ 1$$.(English)Zbl 0573.35030

The authors study the problem $(P)\quad -\Delta u+\gamma (.,u)=f\quad\text{in}\;{\mathcal D}'(G),\quad u=0\quad\text{on}\;\partial G$ where $$G\subset \mathbb R^ N$$ is an open set, $$\gamma (x,s): G\times \mathbb R\to \mathbb R$$ is a function measurable in x and continuous non decreasing in s with $$\gamma (x,0)=0$$ a.e. and $$f\in L^ 1(G)$$. If $$G=\mathbb R^ N$$, $$N\geq 3$$, and the function $$x\to \gamma (x,t)$$ is in $$L^ 1_{\text{loc}}(\mathbb R^ N)$$ for each t, the authors show that (P) has a unique solution in the Marcinkiewicz space. If $$N=2$$ and $$N=1$$, to get the existence of solution they need some additional assumptions on $$\gamma$$. Then, they solve (P) also if $$G$$ is a bounded regular open set. It is interesting to remark that for obtain the existence of solution the classical assumption that $$\gamma$$ is nondecreasing in s can be weakened to $$\gamma (x,s)$$, $$s\geq 0$$.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J61 Semilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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### References:

 [1] Baras, C.R. Acad. Sci. Paris Sér. I 295 pp 519– (1982) [2] Brezis, Variational Inequalities pp 53– (1980) [3] Bénilan, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 pp 523– (1975) [4] DOI: 10.2969/jmsj/02540565 · Zbl 0278.35041
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