Multiple solutions for a critical fractional Kirchhoff type equation with supercritical exponent
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초록
In this paper, we deal with the following fractional Kirchhoff type equation \begin{equation*}\aligned &\ \mathcal{M}\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}} u|^2dx\right)(-\Delta)^su+u\\ =&\ g(x)|u|^{2_{s}^*-2}u+\lambda h(x)|u|^{q-2}u,~\mathrm{in}\ \mathbb{R}^N, \endaligned\end{equation*} where $N>2s$, $\mathcal{M}(t)=a+bt$ with $a\geq 0$ and $b>0$, $s\in (\frac{1}{2},1)$, $h(x)=\frac{|x|^\beta}{1+|x|^\beta}$, $0<\beta<N$ and $2_s^*<q<\frac{2(N+\beta)}{N-2s}$, $2_{s}^*:=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter. By using the Nehari manifold and variational methods, we prove that for $\lambda$ small enough, the number of nontrivial solutions depends on the profile of $g(x)$ for the above equation. Also, for $\lambda$ sufficiently large, we establish the existence of least energy sign-changing solutions. Some novel results improve and generalize the existing ones in the literature.
Abstract
In this paper, we deal with the following fractional Kirchhoff type equation \begin{equation*}\aligned &\ \mathcal{M}\left(\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}} u|^2dx\right)(-\Delta)^su+u\\ =&\ g(x)|u|^{2_{s}^*-2}u+\lambda h(x)|u|^{q-2}u,~\mathrm{in}\ \mathbb{R}^N, \endaligned\end{equation*} where $N>2s$, $\mathcal{M}(t)=a+bt$ with $a\geq 0$ and $b>0$, $s\in (\frac{1}{2},1)$, $h(x)=\frac{|x|^\beta}{1+|x|^\beta}$, $0<\beta<N$ and $2_s^*<q<\frac{2(N+\beta)}{N-2s}$, $2_{s}^*:=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter. By using the Nehari manifold and variational methods, we prove that for $\lambda$ small enough, the number of nontrivial solutions depends on the profile of $g(x)$ for the above equation. Also, for $\lambda$ sufficiently large, we establish the existence of least energy sign-changing solutions. Some novel results improve and generalize the existing ones in the literature.