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### Open Access Journals

DCDS

This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schr*ö*dinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, fluid mechanics, as well as the self-channeling of a high-power ultashort laser in matter. We find the critical exponents for a generalized quasilinear Schrödinger equations and obtain the existence of sign-changing solution with k nodes for any given integer $k ≥ 0$.

CPAA

This paper is concerned with the existence of positive solutions for a class of quasilinear Schrödinger
equations in $R^N$ with critical growth and potential vanishing at infinity. By using a change of variables,
the quasilinear equations are reduced to semilinear one. Since the potential vanish at infinity,
the associated functionals are still not well defined in the usual Sobolev space.
So we have to work in the weighted Sobolev spaces, by Hardy-type inequality,
then the functionals are well defined in the weighted Sobolev space and satisfy
the geometric conditions of the Mountain Pass Theorem. Using this fact,
we obtain a Cerami sequence converging weakly to a solution $v$.
In the proof that $v$ is nontrivial, the main tool is classical arguments
used by H. Brezis and L. Nirenberg in [13].

DCDS

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 14 ], we obtain the existence of positive ground state solutions for the given problem.

*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 [
DCDS

We give an explicit formula for exponential decay properties of
positive solutions for a class of semilinear elliptic equations
with Hardy term in the whole space

*R*^{n}.
DCDS

We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation

$ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}dx} )\Delta u + \lambda V(x)u = f(u),\;x \in {\mathbb{R}^3},$ |

where

>0 is a parameter and the potential

is a nonnegative continuous function with a potential well

which possesses

disjoint bounded components

. Under some conditions imposed on

, multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as

are also studied.

$λ$ |

$V(x)$ |

$Ω: = int(V^{-1}(0))$ |

$k$ |

$Ω_1,Ω_2,···,Ω_k$ |

$f(u)$ |

$λ→ +∞$ |

DCDS-S

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

are constants,

with

is the critical Sobolev exponent in

,

is a potential function on

. Under some more general assumptions on

and

, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.

$ a, \ b>0 $ |

$ 2^*_s = \frac{6}{3-2s} $ |

$ s\in(0, 1) $ |

$ \mathbb{R} ^3 $ |

$ V $ |

$ \mathbb{R} ^3 $ |

$ f $ |

$ V $ |

CPAA

This paper is concerned with a type of quasilinear Schrödinger equations of the form
\begin{eqnarray}
-\Delta u+V(x)u-p\Delta(|u|^{2p})|u|^{2p-2}u=\lambda|u|^{q-2}u+|u|^{2p2^{*}-2}u,
\end{eqnarray}
where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N-2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a sign-changing minimizer with $k$ nodes.

DCDS

In this paper, we consider the following problem
$$
\left\{
\begin{array}{ll}
-\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\
u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N},
\end{array}
\right. (\star)
$$
where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We
prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has
exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by
Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover,
if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a
turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.

DCDS

In this paper, we study the existence of positive
solution for the following p-Laplacain type equations with critical nonlinearity
\begin{equation*}
\left\{
\renewcommand{\arraystretch}{1.25}
\begin{array}{ll}
-\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad
x\in\mathbb{R}^N,\\
u \in \mathcal{D}^{1,p}(\mathbb{R}^N),
\end{array}
\right.
\end{equation*}
where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac
{Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at
infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded,
nonnegative continuous function.
By working in the weighted Sobolev spaces, and using variational method, we
prove that the given problem has at least one positive solution.

DCDS

In this paper, by an approximating argument, we obtain infinitely many radial solutions for the
following elliptic systems with critical Sobolev growth
$$
\left\lbrace\begin{array}{l}
-\Delta u=|u|^{2^*-2}u +
\frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\
-\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v
+ \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\
u = v = 0, \ \ &x \in \partial B, \end{array}\right.
$$
where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.

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