Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity
Yinbin Deng Wei Shuai
Communications on Pure & Applied Analysis 2014, 13(6): 2273-2287 doi: 10.3934/cpaa.2014.13.2273
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].
keywords: critical growth positive solution. Quasilinear Schrödinger equations potential vanishing weighted Sobolev space
Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$
Yinbin Deng Wei Shuai
Discrete & Continuous Dynamical Systems - A 2018, 38(6): 3139-3168 doi: 10.3934/dcds.2018137
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},$
>0 is a parameter and the potential
is a nonnegative continuous function with a potential well
$Ω: = int(V^{-1}(0))$
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
$λ→ +∞$
are also studied.
keywords: Kirchhoff-type equations multiple sign-changing solutions multi-bump solutions concentration behavior
Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents
Yi He Lu Lu Wei Shuai
Communications on Pure & Applied Analysis 2016, 15(1): 103-125 doi: 10.3934/cpaa.2016.15.103
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

keywords: critical growth. Schrödinger-Poisson equation Existence concentration
Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity
Yinbin Deng Yi Li Wei Shuai
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 683-699 doi: 10.3934/dcds.2016.36.683
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.
keywords: p-Laplacain type equations weighted Sobolev space critical growth variational method. vanishing potential

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