## 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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