Journal of Differential Equations最新文献

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation(P_ε)$-\Delta u+\epsilon u=(I_{\alpha }⁎|u{|}^p\left)\right|u{|}^{p-2}u+|u{|}^{q-2}u,\text{in}{R}^N,$where $N\ge 3$ is an integer, $p\in [\frac{N+\alpha }{N},\frac{N+\alpha }{N-2}]$, $q\in (2,\frac{2N}{N-2})$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$ and $\epsilon >0$ is a parameter. We show that as $\epsilon \to 0$ (resp. $\epsilon \to \infty$), the ground state solutions of $\left(P_{\epsilon }\right)$, after appropriate rescalings dependent on parameter regimes, converge in $H^1\left(R^N\right)$ to particular solutions of five different limit equations. We also establish a sharp asymptotic characterisation of such rescalings, and the precise asymptotic behaviour of $u_{\epsilon }\left(0\right)$, ${\Vert \nabla u_{\epsilon }\Vert }_2^2$, ${\Vert u_{\epsilon }\Vert }_2^2$,