arXiv subjects

Zhang Jianjun

Publications and source records attributed to Zhang Jianjun.

3 recordsLinked to original sources

arXiv2026

Stability of the Sobolev--Escobar bridge inequality

We study the local stability of the bridge family \[ \Phi(T):=\inf_{u\in\mathcal A_T}\|\nabla u\|_{L^2(\mathbb R^n_+)}, \qquad T>0,\quad n\ge3, \] where \[ \mathcal A_T := \Bigl\{ u\in \dot H^1(\mathbb R^n_+): \|u\|_{L^{\frac{2n}{n-2}}(\mathbb{R}_{+}^n)}=1,\ \|u\|_{L^{\frac{2(n-1)}{n-2}}(\partial\mathbb{R}_{+}^n)}=T \Bigr\}, \] and $\dot H^1(\mathbb R^n_+)$ is the completion of $C_c^\infty(\overline{\mathbb R^n_+})$ in the norm $\|\nabla \varphi\|_{L^2(\mathbb R^n_+)}$. Let $\mathcal M_T$ denote the set of minimizers of $\Phi(T)$. We prove that, for every $T\neq T_E$, there exists $\alpha_T>0$ such that \[ \|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-\Phi(T)^2 \ge \alpha_T\,d_T(u,\mathcal M_T)^2 +o\!\bigl(d_T(u,\mathcal M_T)^2\bigr) \qquad\text{for all }u\in\mathcal A_T, \] where $T_E$ is the Escobar threshold and $d_T$ is the distance in $\dot H^1(\mathbb R^n_+)$.{for all }u\in\mathcal A_T, \] where $T_E$ is the Escobar threshold and $d_T$ is the distance in $\dot H^1(\mathbb R^n_+)$.

math.AP↗

arXiv2025

Normalized solutions of coupled Sobolev critical Schrodinger equations with mass subcritical couplings

We are concerned with qualitative properties of positive solutions to the following coupled Sobolev critical Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{2^*-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~ \R^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{2^*-2}v+\nu\beta |u|^{\alpha}|v|^{\beta-2}v ~\hbox{in}~ \R^N \end{cases} $$ subject to the mass constraints $\int_{\mathbb{R}^N}|u|^2 \ud x=a^2$ and $\int_{\mathbb{R}^N}|v|^2 \ud x=b^2$, where, $a>0,\,b>0,\,N=3,4$ and $2^*:=\frac{2N}{N-2}$ is the Sobolev critical exponent. The main purpose of this paper is focused on the mass mixed case, i. e., $ \alpha>1,\beta>1,\alpha+\beta<2+\frac{4}{N}$. For some suitable small $\nu>0$, we show that the above system admits two positive solutions, one of which is a local minimizer, and another one is a mountain pass solution. Moreover, as $\nu\to0^+$, asymptotic behaviors of solutions are also considered. Our result gives an affirmative answer to a Soave's type open problem raised by Bartsch {\it et al.} (Calc. Var. Partial Differential Equations 62(1), Paper No. 9, 34, 2023).

math.AP↗

arXiv2025

Multiple normalized solutions for two coupled Gross-Pitaevskii equations with attractive interactions and mass constriants

We are concerned with the following system of two coupled time-independent Gross-Pitaevskii equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~ \R^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{q-2}v+\nu\beta |u|^{\alpha}|v|^{\beta-2}v ~\hbox{in}~ \R^N, \end{cases} $$ which arises in two-components Bose-Einstein condensates and involve attractive Sobolev subcritical or critical interactions, i. e., $\nu>0$ and $\alpha+\beta\leq 2^*$. This system is employed by seeking critical points of the associated variational functional with the constrained mass below $$\int_{\mathbb{R}^N}|u|^2 {\rm d}x=a, \quad \int_{\mathbb{R}^N}|v|^2 {\rm d}x=b.$$ In the mass mixed case, i. e., $2<p<2+\frac{4}{N}<q<2^*$, for some suitable $a,b,\nu$ and $\beta$, the system above admits two positive solutions. In particular, in the case $\alpha+\beta<2^*$, using variational methods on the $L^2$-ball, two positive solutions are obtained, one of which is a local minimizer and the second one is a mountain pass solution.

math.AP↗