## A concentration phenomenon for semilinear elliptic equations.(English)Zbl 1266.35074

In the very interesting paper under review the authors consider the equation $-\Delta u+V(x)u=Q_n(x)|{u}|^{p-2}u$ in a domain $$\Omega \subset \mathbb R^N$$ with zero Dirichlet boundary conditions and where $$p\in(2,2^\ast).$$ It is assumed that $$V\geqq 0$$ and $$Q_n$$ are bounded functions which are positive in a sub-region of $$\Omega$$ and negative outside, and such that the sets $$\{Q_n>0\}$$ shrink to a point $$x_0\in\Omega$$ as $$n\to \infty.$$ The main result of the paper states that if $$u_n$$ is a nontrivial solution corresponding to $$Q_n,$$ then the sequence $$(u_n)$$ concentrates at $$x_0$$ with respect to the $$H^1$$ and certain $$L^q$$-norms. It is also shown that if the sets $$\{Q_n>0\}$$ shrink to two points and $$u_n$$ are ground state solutions, then they concentrate at one of these points.

### MSC:

 35J61 Semilinear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

semilinear elliptic equation; concentration phenomena
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### References:

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