Entire extensions and exponential decay for semilinear elliptic equations

We consider semilinear partial differential equations in $\R^n$ of the form $$\sum_{\frac{|\alpha|}{m}+\frac{|\beta|}{k}\leq 1}c_{\alpha \beta}x^{\beta}D_x^{\alpha}u = F(u)$$ where $k$ and $m$ are given positive integers. Relevant examples are se\-mi\-li\-near Schr\"{o}dinger equations $$-\Delta u +V(x)u= F(u)$$ where the potential $V(x)$ is given by an elliptic polynomial. We propose techniques, based on anisotropic generalizations of the global ellipticity condition of M. Shubin and multiparameter Picard type schemes in spaces of entire functions, which lead to new results for entire extensions and asymptotic behaviour of the solutions. Namely we study solutions (eigenfunctions and homoclinics) in the framework of the Gelfand-Shilov spaces $S^{\mu}_{\nu}(\R^n).$ Critical thresholds are identified for the indices $\mu$ and $\nu$, corresponding to analytic regularity and asymptotic decay, respectively. In the one-dimensional case $-u''+V(x)u=F(u)$ our results for linear equations link up with those given by the classical asymptotic theory and by the theory of ODE in the complex domain, whereas for homoclinics new phenomena concerning analytic extensions are described.