On a class of Kirchhoff problems via local mountain pass
Ambrosio, Vincenzo, 1986- ; Repovš, Dušan, 1954-
In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: ▫$$\begin{cases}-(\varepsilon^2a+\varepsilon b\int ..._{\mathbb{R}^3}|\nabla u|^2 dx) \Delta u + V(x)u = f(u)+\gamma u^5 & \text{in} \; \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \quad u>0 & \text{in} \; \mathbb{R}^3, \end{cases}$$▫ where ▫$\varepsilon>0$▫ is a small parameter, ▫$a,b>0$▫ are constants, ▫$\gamma \in {0,1}$▫, ▫$V$▫ is a continuous positive potential with a local minimum, and ▫$f$▫ is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud. 14 (2014), 483-510; J. Differ. Equ. 252 (2012), 1813-1834; J. Differ. Equ. 253 (2012), 2314-2351).
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