DCDS
Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero
Juntao Sun Jifeng Chu Zhaosheng Feng
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3807-3824 doi: 10.3934/dcds.2013.33.3807
In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum point zero, have at least one and infinitely many homoclinic orbits under certain conditions.
keywords: variational methods Hölder inequality Strongly indefinite problems. Homoclinic orbits Hamiltonian systems asymptotically linear terms
DCDS
Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
Juntao Sun Tsung-Fang Wu Zhaosheng Feng
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 1889-1933 doi: 10.3934/dcds.2018077
We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system:
$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$
where
$\lambda >0$
,
$2 < p \le 4$
and both
$K\left( x\right) $
and
$a\left( x\right) $
are nonnegative functions in
$\mathbb{R}^{3}$
, which satisfy the given conditions, but not require any symmetry property. Assuming that
$% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$
and
$\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$
, we explore the existence of positive solutions, depending on the parameters
$\lambda$
and
$p$
. More importantly, we establish the existence of ground state solutions in the case of
$3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$
.
keywords: Positive solution Schrödinger-Poisson system variational method ground state fibering maps Sobolev embedding theorem concentration-compactness principle

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