Differential and Integral Equations

Critical Schrödinger systems in $\mathbb R^N$ with indefinite weight and Hardy potential

Xuexiu Zhong and Wenming Zou

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By using variational methods, we study the following doubly critical elliptic system: \[ \begin{cases} -\Delta u-\mu_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u= h(x)\alpha|u|^{\alpha-2}|v|^\beta u\quad & \rm{in}\; \mathbb R^N,\\ -\Delta v-\mu_2\frac{u}{|x|^2}-|v|^{2^{*}-2}v= h(x)\beta |u|^{\alpha}|v|^{\beta-2}v\quad & \rm{in}\; \mathbb R^N, \\ \end{cases} \] where $\alpha >1, \beta >1$ satisfying $\alpha+\beta \leq 2^{*}:=\frac{2N}{N-2}\; (N\geq 3)$, and either $\mu_1=\mu_2=0$ or $\mu_1,\mu_2\in (0,\frac{(N-2)^2}{4})$. The function $h(x)$ is allowed to be sign-changing and satisfies weaker conditions which permit the nonlinearities to include a large class of indefinite weights. We obtain the existence of the ground state solutions. The indefiniteness of $h$ makes the problem intrinsically complicated. However, our assumptions on $h$ are almost optimal.

Article information

Differential Integral Equations, Volume 28, Number 1/2 (2015), 119-154.

First available in Project Euclid: 11 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B38: Critical points 35B40: Asymptotic behavior of solutions 35J10: Schrödinger operator [See also 35Pxx] 35J20: Variational methods for second-order elliptic equations


Zhong, Xuexiu; Zou, Wenming. Critical Schrödinger systems in $\mathbb R^N$ with indefinite weight and Hardy potential. Differential Integral Equations 28 (2015), no. 1/2, 119--154. https://projecteuclid.org/euclid.die/1418310424

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