Yanheng Ding

Strongly Indefinite Functionals and Multiple Solutions of Elliptic Systems

We study existence and multiplicity of solutions of the elliptic system \( \left\{{\begin{array}{*{20}l} {- \Updelta u = H_{u} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,} \hfill \\ {- \Updelta v = H_{v} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,\quad u(x) = v(x) = 0\quad {\text{on}}\,\partial \Upomega,} \hfill \\ \end{array}} \right. \) where \( \Upomega \subset {\mathbb{R}}^{N},\,N \ge 3, \) is a smooth bounded domain and \( H \in {\mathcal{C}}^{1} (\overline{\Upomega} \times {\mathbb{R}}^{2},{\mathbb{R}}). \) We assume that the nonlinear term \( H(x,\,u,\,v)\sim \left| u \right|^{p} + \left| v \right|^{q} + R(x,\,u,\,v)\,{\text{with}}\,\mathop {\lim}\limits_{{\left| {(u,v)} \right| \to \infty}} \frac{R(x,\,u,\,v)}{{\left| u \right|^{p} + \left| v \right|^{q}}} = 0, \) where \( p \in (1,\,2^{*}),\,2^{*} : = 2N/(N - 2),\,{\text{and}}\,q \in (1,\,\infty). \) So some supercritical systems are included. Nontrivial solutions are obtained. When H(x, u, v) is even in (u, v), we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if p > 2 (resp. p < 2). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity☆

Abstract In this paper we deal with multiplicity of positive solutions to the p-Laplacian equation of the type − div |∇u| p−2 ∇u =μu q−1 +u p ∗ −1 , u∈W 1,p 0 (Ω), where Ω⊂ R N is a bounded domain, N⩾p2, 2⩽p⩽q ∗ , p ∗ =Np/(N−p) , and μ is a positive parameter. We prove that there exists μ ∗ >0 such that, for each μ∈(0,μ ∗ ) , the equation has at least cat Ω (Ω) positive solutions.

MULTIPLE HOMOCLINICS IN A HAMILTONIAN SYSTEM WITH ASYMPTOTICALLY OR SUPER LINEAR TERMS

This paper is concerned with homoclinic orbits in the Hamiltonian system where H is periodic in t with Hz(t, z) = L(t)z + Rz(t, z), Rz(t, z) = o(|z|) as z → 0. We find a condition on the matrix valued function L to describe the spectrum of operator so that a proper variational formulation is presented. Supposing Rz is asymptotically linear as |z| → ∞ and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where Rz is super linear as |z| → ∞ with assumptions different from those studied previously in relative work, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals in critical point theory.

STATIONARY STATES OF NONLINEAR DIRAC EQUATIONS WITH GENERAL POTENTIALS

We establish the existence of stationary states for the following nonlinear Dirac equation with real matrix potential M(x) and superlinearity g(x,|u|)u both without periodicity assumptions, via variational methods.

Existence and Concentration of Semiclassical Solutions for Dirac Equations with Critical Nonlinearities

We study the semiclassical ground states of the Dirac equation with critical nonlinearity:

Existence and Number of Solutions for a Class of Semilinear Schrödinger Equations

-i\hbar\alpha\cdot\nabla w + a\beta w +V(x)w= W(x)(g(|w|)+|w|)w

A note on superlinear Hamiltonian elliptic systems

for

ON SEMICLASSICAL GROUND STATES OF A NONLINEAR DIRAC EQUATION

x\in\mathbb{R}^3

Periodic Solutions of Hamiltonian Systems

. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. We develop an argument to establish the existence of least energy solutions for

Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system

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