DCDS
Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth
Miaomiao Niu Zhongwei Tang

In this paper, we study a class of nonlinear Schrödinger equations involving the fractional Laplacian and the nonlinearity term with critical Sobolev exponent. We assume that the potential of the equations includes a parameter $λ$. Moreover, the potential behaves like a potential well when the parameter λ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter λ 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 $(-Δ)^{s}$ is nonlocal.

keywords: Nonlinear Schrödinger equation least energy solution critical growth fractional Laplacian
CPAA
Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells
Miaomiao Niu Zhongwei Tang
In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and 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.
keywords: half Laplacian Nonlinear Schrödinger equation least energy solution variational methods.

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