| Abstract: |
| In this talk, we shall discuss the existence of positive solutions and their concentration profile for the following problem:
$\begin{equation*}
-\varepsilon^2\Delta u +c(x)u=b(x) |v|^{q-1}v,\,\,\text{ and } -\varepsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u \quad\text{in } \Omega,
\end{equation*}$
with Neumann boundary data on $\partial\Omega$. The domain is bounded, and the coefficients are considered smooth, positive and bounded. We shall first discuss the existence of positive solutions using some direct method of calculus of variations. Then, we explain the concentration profile of the solutions as the perturbation parameter converges to zero. Our emphasis will be on the dependence of the concentration profile on the coefficients. We conclude by applying the result for different kinds of higher dimensional concentrations for some coupled elliptic systems. |
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