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Precise spectral asymptotics for nonautonomous logistic equations of population dynamics in a ball. (English) Zbl 1128.35078
From the text: We consider the semilinear elliptic eigenvalue problem \(-\Delta u+k(|x|)u^p=\lambda u, u>0\) in \(B_R, u=0\) on \(\partial B_R\), where \(p>1\) is a constant, \(B_R:=\{x\in \mathbb R^N :|x|<R\}\), \( N\geq1\), and \(\lambda>0\) is a parameter. We investigate the global structure of the branch of \((\lambda,u_\lambda)\) of the bifurcation diagram from the point of view of \(L^2\)-theory. To do this, we establish a precise asymptotic formula for \(\lambda=\lambda(\alpha)\) as \(\alpha\to\infty\), where \(\alpha:=\| u_\lambda\|_2\).
MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B32 Bifurcations in context of PDEs
92D25 Population dynamics (general)
35J61 Semilinear elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
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