A concentration behavior for semilinear elliptic systems with indefinite weight
Tóm tắt
Consider the Schrödinger system
$\left\{ \begin{gathered}
- \Delta u + V_{1,n} u = \alpha Q_n (x)|u|^{\alpha - 2} u|v|^\beta , \hfill \\
- \Delta v + V_{2,n} v = \beta Q_n (x)|u|^\alpha |v|^{\beta - 2} v, \hfill \\
u,v \in H_0^1 (\Omega ), \hfill \\
\end{gathered} \right.$
where Ω ⊂ ℝ
N
, α, β > 1, α + β < 2* and the spectrum σ(−Δ + V
i,n
) ⊂ (0, + ∞), i = 1, 2; Q
n
is a bounded function and is positive in a region contained in Ω and negative outside. Moreover, the sets {Q
n
> 0} shrink to a point x
0 ∈ Ω as n → +∞. We obtain the concentration phenomenon. Precisely, we first show that the system has a nontrivial solution (u
n
, v
n
) corresponding to Q
n
, then we prove that the sequences (u
n
) and (v
n
) concentrate at x
0 with respect to the H
1-norm. Moreover, if the sets {Q
n
> 0} shrink to finite points and (u
n
, v
n
) is a ground state solution, then we must have that both u
n
and v
n
concentrate at exactly one of these points. Surprisingly, the concentration of u
n
and v
n
occurs at the same point. Hence, we generalize the results due to Ackermann and Szulkin [Arch. Rational Mech. Anal., 207, 1075–1089 (2013)].
Tài liệu tham khảo
Menyuk, C. R.: Nonlinear pulse propagation in birefringence optical fiber. IEEE J. Quantum Electron, 23, 174–176 (1987)
Pendry, J. B., Schurig, D., Smith, D. R.: Controlling electromagnetic fields. Science, 312(5781), 1780–1782 (2006)
Smith, D. R., Pendry, J. B., Wiltshire, M. C. K.: Metamaterials and negative refractive index. Science, 305(5685), 788–792 (2004)
Veselago, V. G.: The electrodynamics of substances with simultaneously negative values of ɛ and µ., Physics- Uspekhi, 10(4), 509–514 (1968)