DCDS
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
DCDS-S
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*} $
where
$ a, \ b>0 $
are constants,
$ 2^*_s = \frac{6}{3-2s} $
with
$ 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 $
and
$ 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
CPAA
Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $
Lun Guo Wentao Huang Huifang Jia
Communications on Pure & Applied Analysis 2019, 18(4): 1663-1693 doi: 10.3934/cpaa.2019079
We consider the existence of positive solutions for the following fractional Schrödinger-Poisson system
$ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi(x)u = K(x)f(u)+|u|^{2_{s}^{*}-2}u, \ \ & x\in \mathbb{R} ^3, \\ \varepsilon^{2s}(-\Delta)^{s}\phi = u^{2}, \ \ & x \in \mathbb{R} ^3, \end{cases} \end{equation*} $
where
$ s \in (\frac{3}{4}, 1) $
,
$ \varepsilon $
is a small and positive parameter,
$ V $
and
$ K $
are nonnegative potential functions.
$ 2_{s}^{*} $
is the critical exponent with respect to fractional Sobolev embedding theorem. Under some suitable conditions on the nonlinearity
$ f $
and potential functions
$ V $
and
$ K $
, we prove that for
$ \varepsilon $
small, the system has a positive ground state solution concentrating around a concrete set related to
$ V $
and
$ K $
. This result generalizes the result for fractional Schrödinger-Poisson system with subcritical exponent by Yu et al. [39] to critical exponent. Moreover, when
$ V $
attains its minimum and
$ K $
attains its maximum, we also obtain multiple solutions by Ljusternik-Schnirelmann theory.
keywords: Fractional Schrödinger-Poisson system ground state solution concentration behavior multiple solutions

Year of publication

Related Authors

Related Keywords

[Back to Top]