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Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. (English) Zbl 1402.35119

Summary: We are concerned with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation \[\begin{cases} -\left(\epsilon^2 a+b\epsilon \int_{\mathbb{R}^3}|\nabla u|^2\right) \Delta u+M(x)u=\lambda f(u)+|u|^4u,&x\in{\mathbb{R}^3}\\ u \in H^1 (\mathbb{R}^3),~u>0&x\in{\mathbb{R}^3}, \end{cases}\] where \(\epsilon\) is a small parameter, \(a,b>0\) are positive constants and \(\lambda>0\) is a parameter, and \(f\) is a continuous superlinear and subcritical nonlinearity. Suppose that \(M(x)\) has at least one minimum. We first prove that the system has a positive ground state solution \(u_\epsilon\) for sufficiently large \(\lambda>0\) and \(\epsilon>0\) sufficiently small. Then we show that \(u_\epsilon\) converges to the positive ground state solution of the associated limit problem and concentrates to a minimum point of \(M(x)\) in certain sense as \(\epsilon\to 0\). Moreover, some further properties of the ground state solutions are also studied. Finally, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials by minimax theorems and the Ljusternik-Schnirelmann theory.

MSC:

35J60 Nonlinear elliptic equations
35B09 Positive solutions to PDEs
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1235.35093
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