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Standing waves for a class of Kirchhoff type problems in \(\mathbb R^3\) involving critical Sobolev exponents. (English) Zbl 1328.35046

Authors’ abstract: We are concerned with the following Kirchhoff type equation with critical nonlinearity: \[ \begin{cases} -\left( \varepsilon^2 a+\varepsilon b \int_{\mathbb{R}^3} |\nabla u|^2\right) \Delta u+V(x)u=\lambda|u|^{p-2}u+|u|^4u \;\;\text{in }\mathbb{R}^3,\\ u>0, u\in H^1\left(\mathbb{R}^3\right) \end{cases} \] where \(\varepsilon\) is a small positive parameter, \(a, b > 0, \lambda > 0, 2 <p \leq 4\). Under certain assumptions on the potential \(V\), we construct a family of positive solutions \(u_\varepsilon \in H^1\left(\mathbb{R}^3\right)\) which concentrates around a local minimum of \(V\) as \(\varepsilon \to 0\). Although, critical growth Kirchhoff type problem \[ \begin{cases} -\left( \varepsilon^2 a+\varepsilon b \int_{\mathbb{R}^3} |\nabla u|^2\right) \Delta u+V(x)u=f(u)+u^5 \;\;\text{in }\mathbb{R}^3,\\ u>0, u\in H^1\left(\mathbb{R}^3\right) \end{cases} \] has been studied in e.g. [Y. He et al., Adv. Nonlinear Stud. 14, No. 2, 483–510 (2014; Zbl 1305.35033)], where the assumption for \(f(u)\) is that \(f(u) \sim |u|^{p-2}u\) with \(4 < p < 6\) and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. As \(g(u):=\lambda |u|^{p-2}u + |u|^4u\) with \(2 < p \leq 4\) does not satisfy the Ambrosetti-Rabinowitz condition \((\exists \mu >4, 0 < \mu \int^u_0 g(s)ds \leq g(u)u\)), the boundedness of Palais-Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function \(g(s)/s^3\) is not increasing for \(s > 0\) prevents us from using the Nehari manifold directly as usual. Our result extends the main result in [loc. cit.] concerning the existence and concentration of positive solutions to the case where \(f (u) \sim |u|^{p-2}u\) with \(4 < p < 6\).

MSC:

35J60 Nonlinear elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B33 Critical exponents in context of PDEs
35B09 Positive solutions to PDEs

Citations:

Zbl 1305.35033
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