Z. Angew. Math. Phys. (2018) 69:9
2017 Springer International Publishing AG,
part of Springer Nature
published online December 14, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Variational analysis of anisotropic Schr¨odinger equations without
G. A. Afrouzi, M. Mirzapour and Vicent¸iu D. R˘adulescu
Abstract. This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schr¨odinger-type
equations of the form
u = λf (x, u)inΩ,
u =0 on∂Ω,
without the Ambrosetti–Rabinowitz growth condition. Our arguments rely on the existence of a Cerami sequence by using
a variant of the mountain-pass theorem due to Schechter.
Mathematics Subject Classiﬁcation. 35J62, 35J70, 46E35, 58E05.
Keywords. Degenerate anisotropic Sobolev spaces, Variable exponent, Cerami sequence, Mountain-pass theorem.
In quantum mechanics, the Schr¨odinger equation is a partial diﬀerential equation that describes how the
quantum state of a quantum system changes with time. It was formulated in late 1925, and published in
1926, by the Austrian physicist Schr¨odinger . In classical mechanics, Newton’s second law (F = ma)is
used to make a mathematical prediction as to what path a given system will take following a set of known
initial conditions. In quantum mechanics, the analogue of Newton’s law is Schr¨odinger’s equation for a
quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized).
It is not a simple algebraic equation, but in general a linear partial diﬀerential equation, describing the
time-evolution of the system’s wave function (also called a “state function”). The nonlinear Schr¨odinger
equation also describes various phenomena arising in the theory of Heisenberg ferromagnets and magnons,
self-channeling of a high-power ultra-short laser in matter, condensed matter theory, dissipative quantum
mechanics, electromagnetic ﬁelds, plasma physics (e.g., the Kurihara superﬂuid ﬁlm equation). We also
refer to the pioneering paper by Gamow  who was particularly interested in the tunneling eﬀect,which
lead to the construction of the electronic microscope and the correct study of the alpha radioactivity.
The notion of “solution” used by him was not explicitly mentioned in the paper, but it is coherent with
the notion of weak solution introduced several years later by other authors such as Leray, Sobolev and
Schwartz. We refer to Ablowitz et al. , Cazenave , Sulem  for a modern overview and relevant
applications. Recent contributions to the analysis of nonlinear Schr¨odinger equations may be found in
Our main purpose is to consider the nonlinear Schr¨odinger equation in a new setting corresponding
to anisotropic spaces of Sobolev-type. More precisely, the standard linear Laplace operator Δ is replaced