@BF1; @FB; @N; @T], the constrained variation methods were used to find solutions for the equation $$\label{1.4}
\begin{cases}
-\Delta u=\lambda f(u-q),\quad & x\in\Omega,\\
u=0,\quad &x\in\partial\Omega,
\end{cases}$$ under the constraint $\int_\Omega F(u-q)=\mu$, where $\lambda>0$ is a Lagrange multiplier a priori unknown. On the other hand, in [@AS; @AY; @Ni; @Y1; @Y2], the solutions were obtained by using Mountain Pass Lemma for various nonlinearities. For the asymptotic behavior, Berger and Fraenkel [@BF1] began studying the asymptotic behavior for variable $\mu$ and $q$, but the lack of information about $\lambda$ is still an obstacle.

To avoid this obstacle, Yang [@Y1] studied the minimization of the functional $\frac{1}{2}\int_\Omega|\nabla
u|^2-\frac{1}{{\varepsilon}^2}\int_\Omega F(u-q)$ under the natural constraint $\int_\Omega|\nabla u|^2-\frac{1}{{\varepsilon}^2}\int_\Omega uf(u-q)=0$ and obtained the asymptotic behavior of the solutions $u_{\varepsilon}$ as ${\varepsilon}\rightarrow0$ for $\Omega=\mathbb{R}^2_+,~q(x)=Wx_1+d$, where $W,d>0$. That is, set $A_{\varepsilon}=\{x\in\mathbb{R}^2_+: f(u_{\varepsilon}-q)>0\},
\kappa_{\varepsilon}=\frac{1}{{\varepsilon}^2}\int_\Omega f(u_{\varepsilon}-q)$ and $x_{\varepsilon}\in
A_{\varepsilon}$, then $diam A_{\varepsilon}\rightarrow0,~
dist(x_{\varepsilon},\partial\mathbb{R}_+^2)\rightarrow0$ and $\frac{u_{\varepsilon}}{\kappa_{\varepsilon}}-G(x_{\varepsilon},\cdot)\rightarrow0$ in $W_{loc}^{1,r}(\mathbb{R}_+^2)$ for $r\in[1,2)$. Later on, similar results were obtained in [@LYY] for bounded domains with additional information that $q(x_{\varepsilon})\rightarrow\min_\Omega q$. However, it has been pointed out in [@SV] that the solutions obtained above corresponded to desingularization of point-vortex solutions with vanishing vorticity. To get non-vanishing vortex solutions, D. Smets and J. Van Schaftingen [@SV] investigated 