Search

In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying the normalized constraint $\displaystyle\int_{\R^3}u^2=c$, where $a,b,c>0$ are prescribed constants, and the nonlinearities $g(u)$ are very general and of mass super-critical. Under some suitable assumptions on $V(x)$ and $g(u)$, we can prove the existence of ground state normalized solutions $(u_c, \lambda_c)\in H^1(\R^3)\times\mathbb{R}$, for any given $c>0$. Due to the presence of the nonlocal term, the weak limit $u$ of any $(PS)_C$ sequence $\{w_n\}$ may not belong to the corresponding Pohozaev manifold, which is different from the local problem. So we have to overcome some new difficulties to gain the compactness of a $(PS)_C$ sequence., Comment: 21 pages

In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{align*} \left\{ \displaystyle \begin{array}{ll} \displaystyle -\Delta u+ \lambda_1 u =\mu_1\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)u+\beta \left( -\frac{1}{2\pi}\ln(|x|) \ast v^2 \right)u, & x \in ~ \mathbb R^2, \vspace{.4cm}\\ -\Delta v+ \lambda_2 v =\mu_2\left( -\frac{1}{2\pi}\ln(|x|) \ast v^2 \right)v +\beta\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)v, & x \in ~ \mathbb R^2, \end{array} \right.\hspace{1cm} \end{align*} where $\beta, \mu_i, \lambda_i \ (i=1,2)$ are positive constants, $\ast$ denotes the convolution in $\mathbb R^2$. By considering the constraint minimum problem on the Nehari manifold, we prove the existence of ground state solutions for $\beta>0$ large enough. Moreover, we also show that every positive solution is radially symmetric and decays exponentially.

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2, \ \ x\in\Omega, u|_{\partial \Omega }=\frac{\partial u}{\partial n}|_{\partial\Omega}=0, \end{cases} \end{equation*} where $\mu,\lambda,\tau \in \mathbb{R}$, $|\mu|+|\tau|\ne 0$, $\Delta ^2=\Delta \Delta $ denotes the iterated N-dimensional Laplacian, $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial \Omega $, $2^{**}=\frac{2N}{N-4}(N\ge5)$ is the critical Sobolev exponent for the embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{2^{**}}(\Omega)$ and $H_0^2 (\Omega )$ is the closure of $C_0^ \infty (\Omega )$ under the norm $|| u ||:=(\int_{\Omega}|\Delta u|^2)^\frac{1}{2}$. The uncertainty of the sign of $s\log s^2$ in $(0,+\infty)$ has some interest in itself. To know which of the three terms $\mu \Delta u$, $\lambda u$ and $\tau u \log u^2$ has a greater influence on the existence of nontrivial weak solutions, we prove the existence of nontrivial weak solutions to the above problem for $N\ge5$ under some assumptions of $\lambda, \mu$ and $\tau$.

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2} &x\in \Omega, \quad \;\:\, u=0& x\in \partial \Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\R^N$ is a bounded smooth domain, $\lambda, \mu \in \R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(\Omega)\hookrightarrow L^{2^\ast}(\Omega)$. The uncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest in itself. We will show the existence of positive ground state solution which is of mountain pass type provided $\lambda\in \R, \mu>0$ and $N\geq 4$. While the case of $\mu<0$ is thornier. However, for $N=3,4$ $\lambda\in (-\infty, \lambda_1(\Omega))$, we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for $\mu<0$ and $-\frac{(N-2)\mu}{2}+\frac{(N-2)\mu}{2}\log(-\frac{(N-2)\mu}{2})+\lambda-\lambda_1(\Omega)\geq 0$ if $N\geq 3$. Comparing with the results in Br\'ezis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter $\mu$ on logarithmic perturbation is not zero.

In present paper, we study the normalized solutions $(\lambda_c, u_c)\in \R\times H^1(\R^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\R^N}|\nabla u|^2dx\right)\Delta u+\lambda u=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3 $$ satisfying the normalization constraint $ \displaystyle\int_{\R^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c>0$. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as $c\rightarrow 0^+$ as well as $c\rightarrow+\infty$., Comment: 26 pages

In present paper, we study the following Schr\"odinger systems $$\begin{cases} -\Delta u_1+V_1(x)u_1+\lambda_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\R^N,\\ -\Delta u_2+V_2(x)u_2+\lambda_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\R^N,\\ 00, ~2+\frac{4}{N}>r_{1,k}+r_{2,k}, p_i, q_j>2, ~r_{1,k}, r_{2,k}>1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{cases} $$ Under some assumptions on $V_\iota$ $(\iota =1,2)$ and the parameters, we can show the existence of ground state normalized solution $(\lambda_1,\lambda_2; u_1, u_2)\in \R^2 \times H^1(\R^N, \R^2) $ to the above mass sub-critical problem for any given $a_1>0,a_2>0$. Here our nonlinearities are general. The potentials $V_1(x)$ and $V_2(x)$ are also very general such that $\inf ess~\sigma(-\Delta+V_\iota)>-\infty$, which are allowed to be singular at some points., Comment: 43 pages

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 study the following fractional nonlinear Schr\"odinger system $$ \left\{% \begin{array}{ll} (-\Delta)^s u +u=\mu_1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u,~~x\in \R^N,\vspace{2mm}\\ (-\Delta)^s v +v=\mu_2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v,~~x\in \R^N, \end{array}% \right. $$ where $00, \mu_2>0, 12s$, and $\beta \in \R$ is a coupling constant. We investigate the existence and non-degeneracy of proportional positive vector solutions for the above system in some ranges of $\mu_1,\mu_2, p, \beta$. We also prove that the least energy vector solutions must be proportional and unique under some additional assumptions.