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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.