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. Universit√° 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].


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


Zbl 0785.35029