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On positive solutions of semilinear elliptic problems. (English) Zbl 0698.35057
The author solves the Dirichlet problem $(*)\quad -\Delta =f(u)\quad in\quad \Omega,\quad u>0\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega,$ where $$\Omega \subset {\mathbb{R}}^ n$$ is a smooth bounded domain and the nonlinearity f has to “cross” the first eigenvalue of -$$\Delta$$ in $$\Omega$$. Further quite restrictive growth conditions are imposed on f, e.g. $$f^-$$ must be bounded by small constant, which depends on the behaviour of f(t) for $$t\geq 0$$. On the other hand f(0) is allowed to be negative, and the author could treat general elliptic operators instead of -$$\Delta$$.
To solve (*) Leray-Schauder degree and a-priori estimates for solutions of variational inequalities [see also H. Brezis and R. E. L. Turner, Commun. Partial Differ. Equations 2, 601-614 (1977; Zbl 0358.35032)] are used to show first the existence of a nontrivial solution of a corresponding variational inequality. This solution u is shown to be smooth and positive in $$\Omega$$, using the smallness of $$f^-$$. Thus u solves the Dirichlet problem (*).
Reviewer: H.-Ch.Grunau

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 47J05 Equations involving nonlinear operators (general) 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 35J25 Boundary value problems for second-order elliptic equations 35B45 A priori estimates in context of PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000)
##### Keywords:
semilinear equations; positive solution; Dirichlet problem
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