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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

Citations:

Zbl 1266.35074
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