DCDS

In this paper, we consider the following Schrödinger equation with critical growth
$$-\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u \quad \hbox{ in } \mathbb{R}^N, $$ where $N\geq 5$, $2^*$ is the critical Sobolev exponent, $\delta>0$ is a constant, $a(x)\geq 0$ and its zero set is not empty. We will show that if the zero set of $a(x)$ has several isolated connected components
$\Omega_1,\cdots,\Omega_k$ such that the interior of $\Omega_i (i=1, 2, ...,
k)$ is not empty and $\partial\Omega_i (i=1, 2, ..., k)$ is smooth, then for any non-empty
subset $J\subset \{1,2,\cdots,k\}$ and $\lambda$ sufficiently large, the equation admits a solution which is trapped in a
neighborhood of $\bigcup_{j\in J}\Omega_j$. Our strategy to obtain the main results is as follows: By using local mountain pass method combining with penalization of the nonlinearities, we first prove the existence of single-bump solutions which are trapped in the neighborhood of only one isolated component of zero set. Then we construct the multi-bump solution by summing these one-bump solutions as the first approximation solution. The real solution will be obtained by delicate estimates of the error term, this last step is done by using Contraction Image Principle.

CPAA

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.

DCDS

We consider the following two coupled
Schrödinger systems in a bounded domain $\Omega\subset \mathbb{R}^N(N=2,3)$ with Neumann boundary conditions
$$\left\{
\begin{array}{ll}
-\epsilon^2 \triangle u + u
= \mu_1 u^3+ \beta u v^2,\\
-\epsilon^2 \triangle v + v =\mu_2 v^3+ \beta u^2 v,\\
u>0, v>0, \\
\partial u/\partial n = 0,\partial v/\partial n = 0, \mbox{on } \partial \Omega.
\end{array}\right.
$$
Suppose the mean curvature $H(P)$ of the boundary $\partial \Omega$ admits several local maximums( or local minimums), we obtain the existence of segregated solutions $(u_\epsilon,v_\epsilon)$ to the above system such that both of $u_\epsilon$ and $v_\epsilon$ admit more than one local maximums, furthermore as $\epsilon$ goes to zero, the maximum points of $u_\epsilon$ and $v_\epsilon$ concentrate at different local maximum points( or local minimum points) of the mean curvature $H(P)$ respectively.

CPAA

In this paper, we consider the following semilinear Schrödinger equations with ciritical growth
\begin{eqnarray}
-\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u,x\in R^N,
\end{eqnarray}
where $N\geq 4$, $a(x)\geq 0$ and its zero sets are not empty. $2^*$ is the critical Sobolev exponent. $\delta>0$ is a constant such that the operator $-\Delta +\lambda a(x)-\delta$ might be indefinite but is non-degenerate.
We prove the existence of least energy solutions which localize near the potential well
$int \{a^{-1}(0)\}$ for $\lambda$ large enough.

DCDS

We are concerned with the existence of single- and multi-bump solutions
of the equation $-\Delta u+(\lambda a(x)+a_0(x))u=|u|^{p-2}u$, $x\in{\mathbb R}^N$;
here $p>2$, and $p<\frac{2N}{N-2}$ if $N\geq 3$. We require that $a\geq 0$
is in $L^\infty_{loc}({\mathbb R}^N)$ and has a bounded potential well $\Omega$, i.e.
$a(x)=0$ for $x\in\Omega$ and $a(x)>0$ for $x\in{\mathbb R}^N$\$\bar{\Omega}$. Unlike
most other papers on this problem we allow that $a_0\in L^\infty({\mathbb R}^N)$ changes sign. Using variational methods we prove the existence of multibump
solutions $u_\lambda$ which localize, as $\lambda\to\infty$, near prescribed
isolated open subsets $\Omega_1,\dots,\Omega_k\subset\Omega$. The operator
$L_0:=-\Delta+a_0$ may have negative eigenvalues in $\Omega_j$, each bump of
$u_\lambda$ may be sign-changing.

DCDS

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.