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CPAA

In this article, we study the existence and multiplicity of positive
solutions for the Kirchhoff type problem with singular and critical nonlinearities
\begin{eqnarray}
\begin{cases}
-\left(a+b\displaystyle\int_\Omega|\nabla u|^2dx\right)\Delta u=\mu u^{3}+\frac{\lambda}{|x|^{\beta}u^{\gamma}}, &\rm \mathrm{in}\ \ \Omega, \\
u>0, &\rm \mathrm{in}\ \ \Omega, \\
u=0, &\rm \mathrm{on}\ \ \partial\Omega,
\end{cases}
\end{eqnarray}
where $\Omega\subset \mathbb{R}^{4}$ is a bounded smooth domain and $a, b>0$, $\lambda>0$. For all $\mu>0$ and $\gamma\in(0,1)$, $0\leq\beta < 3$, we obtain one positive solution. Particularly, we prove that this problem
has at least two positive solutions for all $\mu> bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.

keywords:
positive solution
,
critical exponent
,
perturbation method.
,
singularity
,
Kirchhoff type problem

CPAA

This paper deals with the existence of multiple positive solutions of a
quasilinear elliptic equation
\begin{eqnarray}
-\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\
u\geq 0, \text{ a.e. }x \in R^N;\\
u \in W^{1,p}(R^N),
\end{eqnarray}
where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is
decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when
parameter $\lambda$ is sufficiently small.

DCDS-B

The existence and multiplicity of homoclinic
orbits for a class of the second order Hamiltonian systems
$\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \forall t \in
\mathbb{R}$, are obtained via the concentration-compactness
principle and the fountain theorem respectively, where $W(t, x)$
is superquadratic and need not satisfy the (

*AR*) condition with respect to the second variable $ x\in\mathbb{R}^{N}$.
CPAA

In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent

$\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u,\ \text{in}\ \mathbb{R}^{3}, \\-Δφ=l(x)u^{2},\ \text{in}\ \mathbb{R}^{3},\end{cases}\end{equation*}$ |

where

and

. Under some appropriate conditions on

and

, we show that there exists

such that the above problem has at least two positive solutions for each

by using the Mountain Pass Theorem and Ekeland's Variational Principle.

$1 < q < 2 $ |

$λ>0 $ |

$ l$ |

$h $ |

$λ^{*}>0 $ |

$λ∈(0,λ^{*}) $ |

DCDS

The existence of weak solutions is obtained for some Kirchhoff type
equations with Dirichlet boundary conditions which are resonant at
an arbitrary eigenvalue under a Landesman-Lazer type condition by
the minimax methods.

keywords:
Resonance
,
critical point
,
Landesman-Lazer
type condition
,
Kirchhoff type problem
,
link.

CPAA

In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems

$ -\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u,\ \ u\in H_{0}.{1}(\Omega ), $ |

where $a, b>0$, $\lambda < a\lambda_1$, $\lambda_1$ is the principal eigenvalue of $(-\triangle, H_0.{1}(\Omega))$. With the help of the Nehari manifold, we obtain that there is $\Lambda>0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 < b < \Lambda$ and $\lambda < a\lambda_1$ and prove that its energy is strictly larger than twice that of ground state solutions. Moreover, we give a convergence property of $u_b$ as $b\searrow 0$. Besides, we firstly establish the nonexistence result of nodal solutions for all $b\geq\Lambda$. This paper can be regarded as the extension and complementary work of W. Shuai (2015)

^{[21]}, X.H. Tang and B.T. Cheng (2016)^{[22]}.
CPAA

In this paper, the existence of a positive solution for the Kirchhoff-type equations in $\mathbb{R}^N$ is proved by using cut-off and monotonicity tricks, which unify and sharply improve the results of Li et al. [Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012) 2285--2294]. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear at infinity.

CPAA

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems
\begin{eqnarray}
\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R,
\end{eqnarray}
where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.

CPAA

In the present paper, the existence and multiplicity of solutions for Kirchhoff type problem involving critical exponent with Dirichlet boundary value conditions are obtained via the variational method.

DCDS

We study the following Kirchhoff type problem:
\begin{equation*}
\left\{
\begin{array}{ccc}
-\left(a+b\int_{\Omega}|\nabla u|^2dx \right) \Delta
u=f(x,u), &\mbox{in} \ \ \Omega, \\
u=0, &\text{on} \ \partial \Omega.
\end{array} \right.
\end{equation*}
Note that $F(x,t)=\int_0^1 f(x,s)ds$ is the primitive function of $f$. In the first result, we prove the existence of solutions by applying the $G-$Linking Theorem when the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_k$ and $\mu_{k+1}$ allowing for resonance with $\mu_{k+1}$ at infinity. In the second result, for the case that the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_1$ and $\mu'_{2}$ allowing for resonance with $\mu'_{2}$ at infinity, we find a nontrivial solution by using the classical Linking Theorem and argument of the characterization of $\mu'_2$. Meanwhile, similar results are obtained for degenerate problem.

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