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Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in \(\mathbb{R} ^2\). (English) Zbl 1429.35081

Summary: In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity \[-\varDelta u+\lambda V(|x|)u+\left(\frac{h^2(|x|)}{|x|^2}+\int\limits^{\infty}_{|x|}\frac{h(s)}{s}u^2(s)ds\right)u=f(u),\,\, x\in\mathbb R^2,\] where \(\lambda > 0, V\) is an external potential and \[h(s)=\frac{1}{2}\int\limits^s_0 ru^2(r)dr=\frac{1}{4\pi}\int\limits_{B_s} u^2(x)dx\] is the so-called Chern-Simons term. Assuming that the external potential \(V\) is nonnegative continuous function with a potential well \(\varOmega \) := int \(V^{-1}(0)\) consisting of \(k + 1\) disjoint components \(\varOmega _0, \varOmega _1, \varOmega _2 \cdots, \varOmega_k\), and the nonlinearity \(f\) has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as \(\lambda \rightarrow +\infty\) are also considered.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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