Solutions to a class of Schrödinger equations (Q2759015)

The author is interested in existence and multiplicity of solutions to the Schrödinger equation NEWLINE\[NEWLINE-i{{d\varphi}/dt}=A\varphi + G_\varphi (x,\varphi)\tag{1}NEWLINE\]NEWLINE of the form \(\varphi(t,x) = e^{i\lambda t}u(x)\), \(u\in H^1({\mathbb R}^3)\), where \(A\) is a Schrödinger operator in \(L^2({\mathbb R}^3)\) and the nonlinearity \(G: {\mathbb R}^3\times {\mathbb C} \to {\mathbb R}\) satisfies \(G(x,0)=0\) and \(G(x,e^{i\theta t}u) = G(x,u)\). Such solutions, when they exist, satisfy NEWLINE\[NEWLINEAu - \lambda u +g(x,u) = 0, \quad u\in H^1({\mathbb R}^3)\tag{S}NEWLINE\]NEWLINE where \(g(x,u) = G_u(x,u)\). The author considers the cases of Schrödinger operators with Rollnik potentials and atomic Hamiltonians. NEWLINENEWLINENEWLINEIn \S\ 2, he treats superlinear nonlinearities using the ``three solution theorem'' of \textit{S. Li} and \textit{M. Willem} [J. Math. Anal. Appl. 189, 6-32 (1995; Zbl 0820.58012)] for existence when \(\lambda \in (\lambda_0, \lambda_e)\), \(\lambda_0 =\inf \sigma(A)\), \(\lambda_e =\inf \sigma_{\text{ess}}(A)\), respectively the spectrum and essential spectrum of \(A\), and Clark's theorem [cf. \textit{P. H. Rabinowitz}, Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] when \(g\) is odd in \(u\) for multiplicity. NEWLINENEWLINENEWLINEIn \S\ 3, he considers the sublinear case, and proves existence as in the former section when \(\lambda \in (\lambda_0, \lambda_e)\) using the Mountain Pass Theorem of \textit{A. Ambrosetti} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)]. He proves also multiplicity when \(g\) is odd in \(u\) using standard arguments (Krasnosel'skij genus and deformation techniques).