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Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$

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  • The paper deals with the following equation of Kirchhoff type, \begin{align*} & -\left ( 1+b\left(\int_{\mathbb{R}^3}|\nabla u|^2dx\right)^r\, \right ) \Delta u+u=k(x)\left (|u|^{q-2}u+\theta g(u)\right )+\lambda h(x)u \end{align*} with $x\in \mathbb{R}^3$, where $u\in H^{1}(\mathbb{R}^3)$, $ b > 0, $ $0 < r < 2, q \in [2(r+1), 6) $, $\theta $ is a small constant, $\lambda$ is a parameter, and a weight function $h (x) \geq 0$. It is known that the linear operator $-\Delta u+u-\lambda h(x)u$ is coercive if $0<\lambda<\lambda_1(h)$ and is non-coercive if $\lambda>\lambda_1(h)$, where $\lambda_1(h)$ is the first eigenvalue of the operator $-\Delta u +u $ with the weight $h(x)$. Under suitable conditions on the functions $k(x)$ and $g(s)$, it is shown that the equation has a positive solution for any $\lambda\in(0,\lambda_1(h))$ and two positive solutions for $\lambda\in(\lambda_1(h), \lambda_1(h) + \tilde \delta )$ with $\tilde \delta > 0$ small. The conditions imposed on $k(x) $ and $g(s)$ are much weaker than those used before, thereby generalizing several existing results on the existence of positive solutions for this type of Kirchhoff equations.
    Mathematics Subject Classification: Primary: 35J20, 35B38; Secondary: 35B09.


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