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Existence and concentration result for the Kirchhoff type equations with general nonlinearities. (English) Zbl 1302.35356

The authors study the existence of positive solutions to the Kirchhoff type equations: \[ -\epsilon^2M\Big(\epsilon^{2-N}\int_{\mathbb R^N}|\nabla u|^2dx\Big)\Delta u +V(x)u=f(u)\quad\text{in }\mathbb R^N,\quad u\in H^1(\mathbb R^N),\quad N\geq 1, \] where \(\epsilon\) is a small parameter, \(M:[0,\infty )\to\mathbb R\), \(f:\mathbb R\to\mathbb R\) and \(V:\mathbb R^N\to\mathbb R\) are given continuous functions. When \(\epsilon=1\), \(M(t)=a+bt\) and \(N=1\), the above equation reduces to the stationary case of the model proposed by Kirchhoff. When \(M(t)\equiv 1\), the above equation reduces to the nonlinear Schrödinger equation. Under suitable conditions on \(M\) and general conditions on \(f\), and employing variational methods, the authors construct a family of positive solutions \((u_\epsilon)_\epsilon\) which concentrates at a local minimum of \(V\) after extracting a subsequence \((\epsilon_k)\).

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B09 Positive solutions to PDEs
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
74K05 Strings
35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
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