On positive solutions of semilinear elliptic problems. (English) Zbl 0698.35057

Quittner, Pavol

Commentat. Math. Univ. Carol. 30, No. 3, 579-585 (1989).

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