Sign changing solutions of nonlinear Schrödinger equations
The authors consider the problem \[ -\Delta u+ b(x)u= f(x, u),\quad x\in\mathbb{R}^n,\quad u\in H^1(\mathbb{R}^n), \] without imposing any symmetry condition on the dependence of \(b(\cdot)\) and \(f(\cdot,u)\) on \(x\). Roughly speaking, \(f\) is assumed to be superlinear and subcritical and to satisfy the sign condition \(u\cdot f(x,u)> 0\), while \(b(x)\to \infty\) as \(|x|\to \infty\). Under such conditions, \textit{P. H. Rabinowitz} [Z. Angew. Math. Phys. 43, No. 2, 270-291 (1992; Zbl 0763.35087)] and the authors themselves [Commun. Partial Differ. Equations 20, No. 9-10, 1725-1741 (1995; Zbl 0837.35043)] proved the existence of a positive and of a negative solution. Subject of the present paper is the existence of a third, a sign changing solution. This result is obtained by approximating the original problem by Dirichlet problems on expanding balls. For the latter the authors base upon previous own results, see e.g. [ibid. 7, No. 1, 115-131 (1996; Zbl 0903.58004)]. Further, it is carefully discussed that the sign changing property is preserved, when passing to the limit.
scientific article; zbMATH DE number 1441647