CPAA
Multiple nontrivial solutions to a $p$-Kirchhoff equation
Anran Li Jiabao Su
Communications on Pure & Applied Analysis 2016, 15(1): 91-102 doi: 10.3934/cpaa.2016.15.91
In this paper, by computing the relevant critical groups, we obtain nontrivial solutions via Morse theory to the nonlocal $p$-Kirchhoff-type quasilinear elliptic equation \begin{eqnarray} (P)\quad\quad &&\displaystyle\bigg[M\bigg(\int_\Omega|\nabla u|^p dx\bigg)\bigg]^{p-1}(-\Delta_pu) = f(x,u), \quad x\in\Omega,\\ && u=0, \quad x\in \partial \Omega, \end{eqnarray} where $\Omega \subset \mathbb R^N$ is a bounded open domain with smooth boundary $\partial \Omega$ and $N \geq 3$.
keywords: critical point local linking. Nonlocal p-Kirchhoff problem Morse theory critical group
DCDS
A bounded resonance problem for semilinear elliptic equations
Jiabao Su Zhaoli Liu
Discrete & Continuous Dynamical Systems - A 2007, 19(2): 431-445 doi: 10.3934/dcds.2007.19.431
In this paper we study the existence and multiplicity of nontrivial solutions for semilinear elliptic resonance problems with a bounded nonlinearity.
keywords: Elliptic equation Morse theory. bounded nonlinearity multiple solutions
DCDS
Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth
Zhaoli Liu Jiabao Su
Discrete & Continuous Dynamical Systems - A 2004, 10(3): 617-634 doi: 10.3934/dcds.2004.10.617
In this paper, existence and multiplicity of nontrivial solutions are obtained for some nonlinear elliptic boundary value problems with perturbation terms of arbitrary growth. Results are obtained via variational arguments.
keywords: minimax methods nontrivial solution. multiplicity perturbation term critical group Elliptic equation resonance
CPAA
Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations
Jiabao Su Rushun Tian
Communications on Pure & Applied Analysis 2010, 9(4): 885-904 doi: 10.3934/cpaa.2010.9.885
We study weighted Sobolev embeddings in radially symmetric function spaces and then investigate the existence of nontrivial radial solutions of inhomogeneous quasilinear elliptic equation with singular potentials and super-$(p, q)$-linear nonlinearity. The model equation is of the form

$ -\Delta_p u+V(|x|)|u|^{q-2}u=Q(|x|)|u|^{s-2}u, x\in R^N,$

$ u(x) \rightarrow 0,$ as $ |x|\rightarrow\infty. $

keywords: Inhomogeneous quasilinear elliptic equation Sobolev type embedding.
CPAA
Resonant problems for fractional Laplacian
Yutong Chen Jiabao Su
Communications on Pure & Applied Analysis 2017, 16(1): 163-188 doi: 10.3934/cpaa.2017008
In this paper we consider the following fractional Laplacian equation
$ \left\{\begin{array}{ll} (-\Delta).s u=g(x, u) & x\in\Omega,\\ u=0, & x \in \mathbb{R}.N\setminus\Omega,\end{array} \right. $
where $ s\in (0, 1)$ is fixed, $\Omega$ is an open bounded set of $\mathbb{R}.N$, $N > 2s$, with smooth boundary, $(-\Delta).s$ is the fractional Laplace operator. By Morse theory we obtain the existence of nontrivial weak solutions when the problem is resonant at both infinity and zero.
keywords: Fractional Laplacian resonance angle condition Palais-Smale condition critical group Morse theory
DCDS
Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities
Mingzheng Sun Jiabao Su Leiga Zhao
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 427-440 doi: 10.3934/dcds.2015.35.427
In this paper, we obtain the existence of infinitely many solutions for the following Schrödinger-Poisson system \begin{equation*} \begin{cases} -\Delta u+a(x)u+ \phi u=k(x)|u|^{q-2}u- h(x)|u|^{p-2}u,\quad &x\in \mathbb{R}^3,\\ -\Delta \phi=u^2,\ \lim_{|x|\to +\infty}\phi(x)=0, &x\in \mathbb{R}^3, \end{cases} \end{equation*} where $1 < q < 2 < p < +\infty$, $a(x)$, $k(x)$ and $h(x)$ are measurable functions satisfying suitable assumptions.
keywords: variational methods. infinitely many solutions Schrödinger-Poisson system concave and convex nonlinearities nonlocal term

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