Positive ground state solutions for a quasilinear elliptic equation with critical exponent
Yinbin Deng Wentao Huang
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4213-4230 doi: 10.3934/dcds.2017179
In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent:
$ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $
which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 < p < 2* = $\frac{{2N}}{{N -2}}$ and V (x) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [14], we obtain the existence of positive ground state solutions for the given problem.
keywords: Ground state solutions quasilinear elliptic equation critical exponent
Least energy solutions for fractional Kirchhoff type equations involving critical growth
Yinbin Deng Wentao Huang
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1929-1954 doi: 10.3934/dcdss.2019126
We study the following fractional Kirchhoff type equation:
$ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $
$ a, \ b>0 $
are constants,
$ 2^*_s = \frac{6}{3-2s} $
$ s\in(0, 1) $
is the critical Sobolev exponent in
$ \mathbb{R} ^3 $
$ V $
is a potential function on
$ \mathbb{R} ^3 $
. Under some more general assumptions on
$ f $
$ V $
, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.
keywords: Fractional Kirchhoff equation Nehari-Pohozaev manifold least energy solutions critical growth

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