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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)