Allegretto, W. Principal eigenvalues for indefinite-weight elliptic problems in \(\mathbb{R}{}^ n\). (English) Zbl 0764.35031 Proc. Am. Math. Soc. 116, No. 3, 701-706 (1992). Summary: We consider the problem \(-\Delta u=\lambda gu\) in \(R^ n\), \(u\to 0\) at \(\infty\) with \(g\) a function that changes sign. Under suitable growth conditions on \(g\) we show that this problem has an eigenvalue \(\lambda\) with a positive solution \(u\), as well as countably many other eigenvalues. Cited in 2 ReviewsCited in 43 Documents MSC: 35J20 Variational methods for second-order elliptic equations 35P15 Estimates of eigenvalues in context of PDEs Keywords:Sobolev’s embedding theorems; eigencurve arguments PDFBibTeX XMLCite \textit{W. Allegretto}, Proc. Am. Math. Soc. 116, No. 3, 701--706 (1992; Zbl 0764.35031) Full Text: DOI