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Solutions of super linear Dirac equations with general potentials. (English) Zbl 1242.47055

Summary: This paper is concerned with solutions to the Dirac equation \[ -i\Sigma \alpha_k \partial_k u + a\beta u + M(x)u = g(x, \| u\| )u. \] Here, \(M(x)\) is a general potential and \(g(x, \| u\| )\) is a self-coupling which grows super-quadratically in \(u\) at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Dirac equation, we construct linking levels of the variational functional \(\Phi_M\) such that the minimax value \(c_M\) based on the linking structure of \(\Phi_M\) satisfies \(0 < c_M < \hat C\), where \(\hat C\) is the least energy of the limit equation. Thus we can show that the \((C)_c\) -condition holds true for all \(c < \hat C\) and consequently we obtain one solution of the Dirac equation.

MSC:

47N50 Applications of operator theory in the physical sciences
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

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