Let Ω ⊂ ℝ2 be a smooth bounded domain and H 1 0 (Ω) be the standard Sobolev space consisting of functions which vanish on ∂Ω and whose gradient is in L2(Ω). In this paper, we investigate critical points of the Trudinger-Moser functional by a heat flow. Our method is based on blow-up analysis, relying on integral estimates.
In this work, we are concerned with inverse nodal problems of Sturm-Liouville operators with eigenparameter dependent boundary conditions on the interval [0, 1]. It is shown that a twin dense subset of the nodal set in a subinterval [a_1,a_2]⊂[0,1] (a_1<12
In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You that the authors constructed the examples whose Lyapunov exponent is discontinuous in the potential with the C0 norm for non-analytic potentials. In this paper, we consider these operators for some Gevrey potential, which is an analytic function with a Gevrey small perturbation on the multi-dimensional torus. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous with the C0 norm in all parameters, such as the energy, the frequency and the potential, for two transformations, the multi-frequency shift and the skew shift. It is complementary to Wang-You’s result.
An odd coloring of a graph is a proper coloring with the additional constraint that each non-isolated vertex has at least one color that appears an odd number of times in its open neighborhood. Determining if a graph is odd k-colorable is NP-complete for k ≥ 3. It is known that planar graphs are odd 8-colorable and triangle-free planar graphs are odd 7-colorable. In this paper we prove that planar graphs without 4-cycles and 5-cycles, or 4-cycles and 6-cycles, are odd 7-colorable.
In this paper, we study the existence of standing wave solutions for the following perturbed quasilinear Schrödinger systems in ℝN {-ε^2 Δ u + V(x)u-ε^2 Δ [(1+u^2)^1/2]u/2(1+u^2)^1/2 =K(x)|u|^2^*-2u+F_u(x,u,v), -ε^2 Δ v + V(x)v-ε^2 Δ [(1+v^2)^1/2]v/2(1+v^2)^1/2 =K(x)|v|^2^*-2v+F_v(x,u,v).. Under some suitable conditions, by using the variational approach, we establish the existence of standing wave solutions for the above system for sufficiently small ε, for any m ∈ ℕ, the system admits at least m pairs of solutions for sufficiently small ε. Moreover, these solutions converge to (uε, vε) → (0, 0) in a suitable Sobolev space as ε → 0. Our results improve and supplement some existing relevant results.
