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### Open Access Journals

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

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

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.

CPAA

We are concerned with standing waves for the following Schrödinger-Poisson equation with critical nonlinearity:
\begin{eqnarray}
&& - {\varepsilon ^2}\Delta u + V(x)u + \psi (x)u = \lambda W(x){\left| u \right|^{p - 2}}u + {\left| u \right|^4}u\;\;{\text{ in }}\mathbb{R}^3, \\
&& - {\varepsilon ^2}\Delta \psi = {u^2}\;\;{\text{ in }}\mathbb{R}^3, u>0, u \in {H^1}(\mathbb{R}^3),
\end{eqnarray}
where $\varepsilon $ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$, $V$ and $W$ are two potentials. Under proper assumptions, we prove that for $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution ${u_\varepsilon }$ by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to [C. Gui, Commun. Partial Differential Equations 21 (1996) 787-820] to show that ${u_\varepsilon }$ concentrates around a set which is related to the set where the potential $V(x)$ attains its global minima or the set where the potential $W(x)$ attains its global maxima as $\varepsilon \to 0$. As far as we know, the existence and concentration behavior of the positive solutions to the Schrödinger-Poisson equation with critical nonlinearity $g(u): = \lambda W(x)|u{|^{p - 2}}u + |u{|^4}u$ $(3

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