On semiclassical states for Dirac equations
This paper is devoted to the study of the existence and concentration of solutions for the stationary Dirac equation \[i\varepsilon\sum_{k=1}^{3}\alpha_{k}\partial_{k}u-a\beta u+V\left( x\right) u=P\left( x\right) f\left( \left\vert u\right\vert \right) u+Q\left( x\right) \left\vert u\right\vert u,\tag{Dirac} \] where \(\varepsilon>0\) is a small parameter, \(a>0\) is a constant, \(V,P,Q\in C\left( \mathbb{R}^{3},\mathbb{R}\right) \) are bounded functions, \(V(x)\) has a negative global minimum and \(P(x)\), \(G(x)\) have a positive global maximum. The aim of this paper is to study how the behavior of the three potentials \(V,P,Q\) affects the existence and concentration (\(\varepsilon\rightarrow0^{+} \)) of semiclassical solutions to (Dirac). The presence of three potentials creates a competition between them: each would try to attract ground states to their minimum and maximum points. First, for any \(\varepsilon>0\) small enough a least energy solution is obtained. Then, the concentration behaviour of these solution as \(\varepsilon\rightarrow0\) is investigated. The concentration position of least solutions is determinated in terms of the potentials \(V,P,Q\): they concentrate at such points \(x\) where \(V(x)\) is small or both \(P(x)\) and \(Q(x)\) are large. Moreover, the exponantial decay estimate of these solutions is deduced.
scientific article; zbMATH DE number 7349839
scientific article