Search

We consider the following critical fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^s u+V(|y'|,y'')u=u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^3\times\mathbb{R}^{N-3}$. If $r^{2s}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, by using a modified finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y'')$. We choose cleverly one of the reduced parameters $\bar{h}$ which depends on the scaling parameter $\lambda$ and avoid to compute the first partial derivative of the reduced functional with respect to $\bar{h}$ directly. Also we have to overcome some difficulties caused by the fractional Laplacian., Comment: 47 pages

In the present paper, we consider the Chern-Simons-Schr\"{o}dinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}\Delta u+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2 u^2,\ \partial_{2}A_{0}=-A_{1}u^{2},\\ &\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial_{1}A_{1}+\partial_{2}A_{2}=0,\\ \end{aligned} \right. \end{equation} where $p>2,$ $\varepsilon>0$ is a parameter and $V:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_{0},A_{1}$ and $A_{2},$ we have to obtain a series of new and technical estimates.

We revisit the following nonlinear critical elliptic equation \begin{equation*} -\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about the periodicity of bubbling solutions. Here we try to investigate some related problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely many bubbling solutions of the problem above. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the potential function $Q(y),$ i.e. the bubbling solution whose blow-up set is $\{(jL,0,...,0):j=0,\pm 1, \pm 2,..., \pm m\}$ must be periodic in $y_{1}$ provided that $L$ is large enough, where $m$ is the number of the bubbles which is large enough but independent of $L.$ Moreover, we also show a non-existence of this bubbling solutions for the problem above if the local minimum of $Q(y)$ does not equal to zero., Comment: There is a obstacle we ignored before and we can not overcome

We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schr\"odinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points. Precisely, applying the finite dimensional reduction method, we first obtain the existence of $k$-peak concentrated solutions and especially describe the relationship between the chemical potential $\mu$ and the attractive interaction $a$. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the $k$-peak solutions with prescribed $L^2$-norm, by use of the local Pohozaev identities, the blow-up analysis and the maximum principle associated to the nonlocal operator $(-\Delta)^s$. To our best knowledge, there is few results on the excited normalized solutions of the fractional Schr\"odinger equations before this present work. The main difficulty lies in the non-local property of the operator $(-\Delta)^s$. First, it makes the standard comparison argument in the ODE theory invalid to use in our analysis. Second, because of the algebraic decay involving the approximate solutions, the estimates, on the Lagrange multiplier for example, would become more subtle. Moreover,when studying the corresponding harmonic extension problem, several local Pohozaev identities are constructed and we have to estimate several kinds of integrals that never appear in the classic local Schr\"odinger problems. In addition, throughout our discussion, we need to distinct the different cases of $p$, which are called respectively that the mass-subcritical, the mass-critical, and the mass-supercritical case, due to the mass-constraint condition. Another difficulty comes from the influence of the different degenerate rates along different directions at the critical points of the potential.