Existence and convergence of sign-changing solutions for Kirchhoff-type $p$-Laplacian problems involving critical exponent in $\mathbb{R}^N$

Abstract

We investigate the existence of sign-changing solutions for Kirchhoff-type problems with $p$-Laplacian involving critical exponent: \begin{equation*} -\left(1+b \vert \nabla v \vert_p^p\right)\Delta_p v+a(x)\vert v\vert^{p-2}v=\vert v\vert^{p^*-2}v+\lambda f(v), \quad x\in \mathbb{R}^N, \end{equation*} where $b$ and $\lambda$ are positive parameters, $\Delta_p v= \operatorname{div}(\vert \nabla v\vert^{p-2} \nabla v)$, $p^*=\frac{Np}{N-p}$, $1<p<N$, and $\vert \cdot \vert_p$ is the Lebesgue $L^p$-norm. For sufficiently large $\lambda$, employing minimization techniques, quantitative deformation lemma and the constrained variational method, we demonstrate the existence of a least-energy sign-changing solution, whose energy is greater than twice that of the ground state solution. Additionally, we show the convergence behavior of the solution as the parameter $b\searrow0$. Our findings generalize and extend upon recent results in the literature.

Keywords

• sign-changing solution
• variational methods
• $p$-laplacian
• kirchhoff-type problem