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Concentration for a biharmonic Schrödinger equation. (English) Zbl 1373.35107

The paper deals with the nonlinear stationary biharmonic Schrödinger equation \[ \begin{cases} \varepsilon^4 \Delta^2u+V(x)u=P(x)f(|u|)u, & x\in \mathbb{R}^N,\\ u(x)\to 0, & \text{as}\;|x|\to\infty, \end{cases} \] where \(\varepsilon\) stands for the Planck constant, and \(V\) and \(P\) are spatial distributions of external potentials. By means of variational methods, the author describes concentration phenomena for the solutions when \(\varepsilon\to0.\)

MSC:

35J10 Schrödinger operator, Schrödinger equation
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35J35 Variational methods for higher-order elliptic equations
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