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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.

In this paper, we investigate the following critical elliptic equation $$ -\Delta u+V(y)u=u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,\R^{N},\,\,u\in H^{1}(\R^{N}), $$ where $V(y)$ is a bounded non-negative function in $\R^{N}.$ Assuming that $V(y)=V(|\hat{y}|,y^{*}),y=(\hat{y},y^{*})\in \R^{4}\times \R^{N-4}$ and gluing together bubbles with different concentration rates, we obtain new solutions provided that $N\geq 7,$ whose concentrating points are close to the point $(r_{0},y^{*}_{0})$ which is a stable critical point of the function $r^{2}V(r,y^{*})$ satisfying $r_{0}>0$ and $V(r_{0},y^{*}_{0})>0.$ In order to construct such new bubble solutions for the above problem, we first prove a non-degenerate result for the positive multi-bubbling solutions constructed in \cite{PWY-18-JFA} by some local Pohozaev identities, which is of great interest independently. Moreover, we give an example which satisfies the assumptions we impose., Comment: 36pages

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schr\"odinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schr\"odinger-Newton system. Precisely, we give the precise description of the chemical potential $\mu$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schr\"odinger-Newton system, which is markedly different from the corresponding Schr\"odinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schr\"odinger-Newton system., Comment: Accepted by Ann. Sc. Norm. Super. Pisa Cl. Sci. arXiv admin note: text overlap with arXiv:1909.08828

This paper deals with the following nonlinear perturbed fractional Laplacian equation $$(-\Delta)^s u = K(|y'|,y'')u^{\frac{N+2s}{N-2s}\pm\epsilon},\,\,u>0,\,\,u\in D^{1,s}(\mathbb{R}^N),$$ where $00$ is a small parameter and $K(y)$ is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\geq 4,\max\{\frac{N+1-\sqrt{N^{2}-2N+9}}{4},\frac{3-\sqrt{N^{2}-6N+13}}{2}\}0$ and $K(r_0, y_0'')>0,$ then the above problem has large number of bubble solutions if $\epsilon>0$ is small enough. Also there exist solutions whose functional energy is in the order $\epsilon^{-\frac{N-2s-2}{(N-2s)^{2}}}$. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bubble solutions. Moreover, the concentration points of the bubble solutions include a saddle point of $K(y)$., Comment: 44pages

In this paper, we consider the following weakly coupled nonlinear Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} -\epsilon^{2}\Delta u_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + \beta|u_1|^{p - 2}|u_2|^pu_1, & x\in \mathbb{R}^N,\\ -\epsilon^{2}\Delta u_2 + V_2(x)u_2 = |u_2|^{2p - 2}u_2 + \beta|u_2|^{p - 2}|u_1|^pu_2, & x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $\epsilon>0$, $\beta\in\mathbb{R}$ is a coupling constant, $2p\in (2,2^*)$ with $2^* = \frac{2N}{N - 2}$ if $N\geq 3$ and $+\infty$ if $N = 1,2$, $V_1$ and $V_2$ belong to $C(\mathbb{R}^N,[0,\infty))$. When $p\ge 2$ and $\beta>0$ is suitably small, we show that the problem has a family of nonstandard solutions $\{w_{\epsilon} = (u^1_{\epsilon},u^2_{\epsilon}):0<\epsilon<\epsilon_{0}\}$ concentrating synchronously at the common local minimum of $V_1$ and $V_2$. All decay rates of $V_i(i=1,2)$ are admissible and we can allow that $\beta>0$ is close to $0$ in this paper. Moreover, the location of concentration points is given by local Pohozaev identities. Our proofs are based on variational methods and the penalized technique., Comment: 34 pages

In the present paper, we consider the nonlocal Kirchhoff problem \begin{eqnarray*} -\left(\epsilon^2a+\epsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $10$ is a parameter. Under some mild assumptions on the function $V$, we obtain multi-peak solutions for $\epsilon$ sufficiently small by Lyapunov-Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of elliptic singular perturbation problems.

In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $10$ sufficiently small, under some mild assumptions on the potential function $V:\mathbb{R}^3\to \mathbb{R}$. The existence result is obtained by applying the Lyapunov-Schmidt reduction method. It seems to be the first time to study singularly perturbed Kirchhoff problems by reduction method, as all the previous results were obtained by various variational methods. Another advantage of this approach is that it gives a unified proof to the perturbation problem for all $p\in (1,5)$, which is quite different from using variational methods in the literature. The local uniqueness result is totally new. It is obtained by using a type of local Pohozaev identity, which is developed quite recently by Deng, Lin and Yan in their work "On the prescribed scalar curvature problem in $\mathbb{R}^N$, local uniqueness and periodicity." (see J. Math. Pures Appl. (9) {\bf 104}(2015), 1013-1044)., Comment: arXiv admin note: text overlap with arXiv:1610.07807

We are concerned with the following Schr\"odinger-Newton problem \begin{equation} -\varepsilon^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(\xi)}{|x-\xi|}d\xi\big)u,~x\in \mathbb R^3. \end{equation} For $\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schr\"odinger-Newton problem is quite different from those of Schr\"odinger equations.

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-\Delta)^{s} u-\ds\frac{\mu u}{|x|^{2s}}=|u|^{2^*_s-2}u+au, & \hbox{$\text{in}~ \Omega$},\vspace{0.1cm} u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array}% \right. \end{equation*} provided $N>6s$, $\mu\geq0$, $0< s<1$, $2^*_s=\frac{2N}{N-2s}$, $a>0$ is a constant and $\Omega$ is an open bounded domain in $\R^N$ which contains the origin.

In this paper, we investigate the following elliptic problem involving double critical Hardy-Sobolev-Maz'ya terms: $$ \left\{\begin{array}{ll} -\Delta u = \mu\frac{|u|^{2^*(t)-2}u}{|y|^t} + \frac{|u|^{2^*(s)-2}u}{|y|^s} + a(x) u, & {\rm in}\ \Omega,\\ \quad u = 0, \,\, &{\rm on}\ \partial \Omega, \end{array} \right. $$ where $\mu\geq0$, $a(x)>0$, $2^*(t)=\frac{2(N-t)}{N-2}$, $2^*(s) = \frac{2(N-s)}{N-2}$, $0\leq t6+t$ when $\mu>0,$ and $N>6+s$ when $\mu=0,$ and $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate Morse indices of these nodal solution., Comment: 11pages

Let $1p^{2}+p,a(0)>0$ and $\Omega$ satisfies some geometry conditions if $0\in\partial\Omega$, say, all the principle curvatures of $\partial\Omega$ at $0$ are negative, then the above problem has infinitely many solutions., Comment: 28pages,no figure

In this paper, we study the nonlinear Schr\"{o}dinger equation with non-symmetric electromagnetic fields $$\Big(\frac{\nabla}{i}-A_{\epsilon} x)\Big)^2 u+V_{\epsilon}(x)u=f(u),\ u\in H^1 (\mathbb{R}^N,\mathbb{C}), $$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions and $f(u)$ is a nonlinearity satisfying some nondegeneracy condition. Applying localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions., Comment: 39fages, 0 figures. arXiv admin note: text overlap with arXiv:1210.8209, arXiv:1209.2824 by other authors