Rother, W. Nonlinear scalar field equations. (English) Zbl 0755.35082 Differ. Integral Equ. 5, No. 4, 777-792 (1992). Some existence results are proved for the nonlinear eigenvalue problem \[ -\Delta u-q(x)| u|^{\sigma_ 1}u+r(x)| u|^{\sigma_ 2}u=\lambda u \] in \(\mathbb{R}^ N\), where \(N\geq 3\), \(0<\sigma_ 1<4/(N-2)\), \(\sigma_ 2\geq 4/(N-2)\). Some conditions for \(q\) and \(r\) are formulated under which the solutions decay exponentially. Reviewer: St.Tersian (Russe) Cited in 1 ReviewCited in 4 Documents MSC: 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J20 Variational methods for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) Keywords:exponential decay of solutions; sub-and supercritical exponent PDFBibTeX XMLCite \textit{W. Rother}, Differ. Integral Equ. 5, No. 4, 777--792 (1992; Zbl 0755.35082)