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Existence of positive solutions for semilinear elliptic equations in general domains. (English) Zbl 0664.35029
The authors consider the existence of positive solutions to the Dirichlet problem \[ (1)\quad \Delta u(x)+f(u(x))=0,\quad x\in \Omega;\quad u(x)=0,\quad x\in \partial \Omega, \] where \(\Omega\) is a bounded domain in \(R^ n\) with smooth boundary and f is a continuous function on R. For it, they introduce the notion of “eccentricity”, e(\(\Omega)\), of a domain \(\Omega\) with the property that \(1\leq e(\Omega)<+\infty\) and \(e(\Omega)=1\) if and only if \(\Omega\) is a ball, and the notion of the “nonlinearity” of f, N(f) (for instance, if f is a linear function, \(N(f)=1\); if \(f(u)=u^ k\), \(k<1\), \(N(f)=\infty\) and if \(f(u)=u^ k\), \(k>1\), \(N(f)<1)\). They prove, using a variation of the method of upper and lower solutions, that if \(N(f)>e(\Omega)\) then there exist positive solutions to (1) on all domains \(\lambda\) \(\Omega\) if \(\lambda\) is sufficiently large (equivalently, positive solutions to the Dirichlet problem for \(\Delta u+\mu f(u)=0\) exist on \(\Omega\) for some range of \(\mu)\). Then, they give some applications of this result.
Also, they consider in more detail the case where \(\Omega\) is an n-ball and the Neumann problem on n-balls.
The used method may be applied to more general elliptic operators.
Reviewer: A.Cañada

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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