Abstract
In this paper, we study a class of nonlinear Schrödinger equations involving the half Laplacian and critical growth. We assume that the potential of the nonlinear Schrödinger equation includes a parameter \({\lambda}\). Moreover, the potential behaves like a potential well when the parameter \({\lambda}\) is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, for \({\lambda}\) large, localizes near the bottom of the potential well. Moreover, if the zero set int \({V^{-1}(0)}\) of \({V(x)}\) includes more than one isolated component, then \({u_{\lambda}(x)}\) will be trapped around all the isolated components. However, in Laplacian case when \({s = 1}\), for \({\lambda}\) large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int \({V^{-1}(0)}\). This is the essential difference with the Laplacian problems since the operator \({(- \Delta)^{1/2}}\) is nonlocal.
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Niu, M., Tang, Z. Least energy solutions for nonlinear Schrödinger equations involving the half Laplacian and critical growth. J. Fixed Point Theory Appl. 18, 367–395 (2016). https://doi.org/10.1007/s11784-016-0286-z
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DOI: https://doi.org/10.1007/s11784-016-0286-z