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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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