## Multi-bump solutions for a semilinear Schrödinger equation.(English)Zbl 1187.35239

Authors’ abstract: We study the existence of multi-bump solutions for the semilinear Schrödinger equation
$-\Delta u+(1+ \varepsilon a(x))u= |u|^{p-2}u, \quad u\in H^1(\mathbb R^N),$
where $$N\geq 1$$, $$2<p<2N/(N-2)$$ if $$N\geq 3$$, $$p>2$$ if $$N=1$$ or $$N=2$$, and $$\varepsilon>0$$ is a parameter. The function $$a$$ is assumed to satisfy the following conditions: $$a\in C(\mathbb R^N)$$, $$a(x)>0$$ in $$\mathbb R^N$$, $$a(x)=o(1)$$ and $$\ln(a(x))= o(|x|)$$ as $$|x|\to\infty$$. For any positive integer $$n$$, we prove that there exists $$\varepsilon(n)>0$$ such that, for $$0<\varepsilon< \varepsilon(n)$$, the equation has an $$n$$-bump positive solution. Therefore, the equation has more and more multibump solutions as $$\varepsilon\to 0$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 35B09 Positive solutions to PDEs
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