## Some $$L^ 1$$ existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions.(English)Zbl 0652.35043

The authors study semilinear elliptic boundary value problems of the general type $f\in -\Delta u+\beta (u)\quad on\quad \Omega;\quad 0\in \partial u/\partial \nu +\gamma (u)\quad on\quad \partial \Omega$ where $$\beta$$ and $$\gamma$$ are maximal monotone graphs with $$D(\beta)\cap \gamma^{-1}(0)\neq \emptyset$$ and $$f\in L_ 1(\Omega)$$. A triple $$(u,v,w)\in W_{1,1}(\Omega)\times L_ 1(\Omega)\times L_ 1(\partial \Omega)$$ is called a solution, if v(x)$$\in \beta (u(x))$$ a.e. on $$\Omega$$, w(x)$$\in \gamma (u(x))$$ a.e. on $$\partial \Omega$$ and $\int_{\Omega}v\phi +\int_{\Omega}\nabla u\nabla \phi +\int_{\partial \Omega}v\phi =\int_{\Omega}f\phi \quad for\quad all\quad \gamma \in W_{1,\infty}(\Omega).$ They show that existence of a solution implies $$\int_{\Omega}f\in range(B)$$ with $$B=| \partial \Omega | \gamma +| \Omega | \beta,$$ while conversely $$\int_{\Omega}\in int(range(B))$$ implies existence - the borderline cases $$\int_{\Omega}f=\sup range(B)$$ (resp. inf) are characterized, too. The basis is an approximation theorem for such solutions which has its own interest. From the vast literature concerning these problems let us just recall H. Brezis [J. Math. Pures Appl., IX. Ser. 51, 1-168 (1972; Zbl 0237.35001)].
Reviewer: M.Wiegner

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 47H05 Monotone operators and generalizations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)

### Keywords:

semilinear; maximal monotone graphs; existence; approximation

Zbl 0237.35001
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### References:

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