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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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