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Asymptotic theory of singular semilinear elliptic equations. (English) Zbl 0589.35046

The authors establish necessary and sufficient conditions for the existence of two positive solutions of the semilinear elliptic equation \(\Delta u+q(| x|)u=f(x,u)\) in an exterior domain \(\Omega \subset {\mathbb{R}}^ n\), \(n\geq 1\), where q, f are real-valued and locally Hölder continuous, and f(x,u) is nonincreasing in u for each fixed \(x\in \Omega\). The proofs are based on first establishing a necessary and sufficient condition for the existence of positive solutions for a nonlinear ordinary differential equation. The criteria are then applied to the radial majorants of the equation considered to construct suitable super- and subsolutions.
A prototype is the equation \(\Delta\) u-\(k^ 2u=p(x)u^{-\lambda}\), \(k>0\), \(\lambda >0\). Let \(p^*(t)=\max_{| x| =t}p(x)\). Their Corollary 3.6 states that if \(\int^{\infty}t^ 0e^{-\sigma t}p^*(t)dt<\infty\), \(\rho =(n-1)(\lambda +1)/2,\quad \sigma =K(\lambda +1)\) then the equation has a positive solution u(x)\(\leq K| x|^{(1-n)/2}e^{K(x)}\) for some \(K>0\), in some exterior domain \(\Omega_ T=\{x:\) \(| x| \geq T>0\}\). If \(\int^{\infty}t^{\rho}e^{\sigma t}p^ x(t)dt<\infty\) the equation has two positive solutions \(u_ 1\) and \(u_ 2\) in \(\Omega_ T\) for which both \(| x|^ me^{k| x|}u_ 1(x)\) and \(| x|^ me^{-k(x)}u_ 2(x)\), \(m=(n-1)/2,\) are bounded and bounded away from zero in \(\Omega_ T\).

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
34B15 Nonlinear boundary value problems for ordinary differential equations
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