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Nonlinear scalar field equations. (English) Zbl 0755.35082
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)

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)
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