A concentration phenomenon for $$p$$-Laplacian equation.(English)Zbl 1437.35414

Summary: It is proved that if the bounded function of coefficient $$Q_n$$ in the following equation $$-\operatorname{div}\{ |\nabla u|^{p-2} \nabla u\} + V(x) |u|^{p-2} u= Q_n (x) |u|^{q-2}u$$ as $$x\in \partial \Omega$$. $$u(x)\to 0$$ as $$|x|\to \infty$$ is positive in a region contained in $$\Omega$$ and negative outside the region, the sets $$\{Q_n > 0 \}$$ shrink to a point $$x_0 \in \Omega$$ as $$n \to\infty$$, and then the sequence $$u_n$$ generated by the nontrivial solution of the same equation, corresponding to $$Q_n$$, will concentrate at $$x_0$$ with respect to $$W_0^{1, p}(\Omega)$$ and certain $$L^s(\Omega)$$-norms. In addition, if the sets $$\{Q_n > 0 \}$$ shrink to finite points, the corresponding ground states $$\{u_n \}$$ only concentrate at one of these points. These conclusions extend the results proved in the work of N. Ackermann and A. Szulkin [Arch. Ration. Mech. Anal. 207, No. 3, 1075–1089 (2013; Zbl 1266.35074)] for case $$p = 2$$.

MSC:

 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations

Zbl 1266.35074
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