## A note on a semilinear elliptic equation on $$\mathbb{R}^ n$$.(English)Zbl 0836.35045

Ambrosetti, A. (ed.) et al., Nonlinear analysis. A tribute in honour of Giovanni Prodi. Pisa: Scuola Normale Superiore, Quaderni. Universitá di Pisa. 307-317 (1991).
The author proves the existence of at least one positive and one negative nontrivial solution in $$W^{1,2} (\Omega, \mathbb{R})$$ to the equation $\sum^n_{i,j = 1} \bigl( a_{ij} (x) u_{x_j} \bigr)_{x_i} + a(x)u = f(x,u),\;x \in \Omega, \tag{1}$ where $$\Omega = \mathbb{R}^n$$ or $$\Omega = \mathbb{R} \times O$$, $$O \subset \mathbb{R}^{n - 1}$$ a bounded domain, and the coefficients and the nonlinearity in (1) are periodic in $$x_1, \dots, x_n$$ (if $$\Omega = \mathbb{R}^n)$$ or periodic in $$x_1$$ (if $$\Omega = \mathbb{R} \times O)$$. Meanwhile, this result has been improved in the paper [P. H. Rabinowitz and V. C. Zelati, Commun. Pure Appl. Math. 45, No. 10, 1217-1269 (1992; Zbl 0785.35029)], where infinitely many solutions to (1) are obtained.
For the entire collection see [Zbl 0830.00011].

### MSC:

 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations

Zbl 0785.35029