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 \(\gamma (x,s)\), \(s\geq 0\).


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)
Full Text: DOI


[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 · doi:10.2969/jmsj/02540565
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.