Multi-peak solutions to Kirchhoff equations in \(\mathbb{R}^3\) with general nonlinearity. (English) Zbl 1398.35058
Summary: We are concerned with the existence of multi-peak solutions to Kirchhoff equation
\[
-\Big(\epsilon^2 a + \epsilon b \mathop{\int}\limits_{\mathbb{R}^3} | \nabla u |^2 d x\Big) \Delta u + V(x) u = f(u),\quad x \in \mathbb{R}^3,\eqno{(0.1)}
\]
where \(\epsilon > 0\) is a small parameter, \(a\), \(b > 0\) are constants. Under general conditions of \(f\), we construct a family positive solutions \(u_\epsilon \in H^1(\mathbb{R}^3)\) which concentrates around the isolated components of positive local minima of \(V\) as \(\epsilon \rightarrow 0^+\). Our result generalize the previous results on single peak solutions to multi-peak solutions.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J50 | Variational methods for elliptic systems |
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