## 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

Zbl 1305.35033
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### References:

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