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In this paper, we prove the multiplicity of nontrivial solutions for a class of fractional-order elliptic equation with magnetic field. Under appropriate assumptions, firstly, we prove that the system has at least two different solutions by applying the mountain pass theorem and Ekeland's variational principle. Secondly, we prove that these two solutions converge to the two solutions of the limit problem. Finally, we prove the existence of infinitely many solutions for the system and its limit problems, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of periodic solutions for a class of delay Hamiltonian systems on time scales with impulsive effects. The variational principle is given, and some existence theorems and two multiplicity results of periodic solutions are obtained. Finally, one example is presented to illustrate the feasibility and effectiveness of our results. Our results are new even in both the differential equations case and the difference equations case. [ABSTRACT FROM AUTHOR]

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian system on time scale $\mathbb{T}$ where u( t) denotes the delta (or Hilger) derivative of u at t, $u^{\Delta^{2}}(t)=(u^{\Delta})^{\Delta}(t)$, σ is the forward jump operator, T is a positive constant, A( t)=[ d( t)] is a symmetric N× N matrix-valued function defined on $[0,T]_{\mathbb{T}}$ with $d_{ij}\in L^{\infty}([0,T]_{\mathbb{T}},\mathbb{R})$ for all i, j=1,2,..., N, and $F:[0,T]_{\mathbb{T}}\times \mathbb{R}^{N}\rightarrow\mathbb{R}$. By establishing a proper variational setting, two existence results and two multiplicity results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the existence and multiplicity of solutions for some Dirichlet impulsive problems and some existence and multiplicity results are obtained. The solutions are sought by means of Lax–Milgram theorem and some critical theorems. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems. [Copyright &y& Elsevier]