Perturbed elliptic equations with oscillatory nonlinearities
Zuji Guo Zhaoli Liu
Discrete & Continuous Dynamical Systems - A 2012, 32(10): 3567-3585 doi: 10.3934/dcds.2012.32.3567
In this paper, arbitrarily many solutions, in particular arbitrarily many nodal solutions, are proved to exist for perturbed elliptic equations of the form \begin{equation*}\label{} \left\{ \begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N, \\ u\in W^{1,p}(\mathbb R^N), \end{array} \right. (P_\varepsilon) \end{equation*} where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb R)$ oscillates either near the origin or near the infinity, and $\epsilon$ is a real number. For $g$ it is only required that $g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on $Q$ and $f$ the following results which are special cases of more general ones are proved: the unperturbed problem $(P_0)$ has infinitely many nodal solutions, and for any $n\in\mathbb N$ the perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions provided that $|\epsilon|$ is sufficiently small.
keywords: invariant sets of desending flow Arbitrarily many solutions perturbed elliptic equations nodal solutions p-Laplacian operator. osillatory terms
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
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
Positive solutions of a nonlinear Schrödinger system with nonconstant potentials
Haidong Liu Zhaoli Liu
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1431-1464 doi: 10.3934/dcds.2016.36.1431
Existence of a solution of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{aligned} & - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\ & - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\ & u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of $x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are invariant under the action of a finite subgroup of $O(N)$ or there is no such invariance assumption. In either case the result is obtained both for $\beta$ small and for $\beta$ large in terms of $V_j$ and $\mu_j$.
keywords: variational methods. Nonlinear Schrödinger system

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