We consider the quasilinear Schrödinger equation involving a general nonlinearity at critical growth. By using Jeanjean’s monotonicity trick and the Pohozaev identity we get the existence results that generalize an earlier work [H. Liu and L. Zhao, Existence results for quasilinear Schrödinger equations with a general nonlinearity, Commun. Pure Appl. Anal., 19(6):3429–3444, 2020] about the subcritical case to the critical case.
We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurinčikas in a problem session of a recent workshop on universality.
In this paper, we answer the questions posed by Gundersen and Yang about the entire solutions of a class of nonlinear homogeneous binomial differential equations and obtain explicit forms of all the entire solutions of this type of differential equations. Moreover, we provide some examples to demonstrate that the equation solutions we obtained are accurate.
We obtain an asymptotic formula for the smoothly weighted first moment of quadratic Dirichlet L-functions at central values, with explicit main terms and an error term that is “square-root” of the main term.
Let λ2, λ3, λ4, λ5 be nonzero real numbers, not all negative. Let (mathfrak{V}) be a well-spaced sequence. Assume that λ2/λ3 is irrational and algebraic, and δ > 0. Let (Eleft(mathfrak{V},N,delta right)) be the number of (upsilon in mathfrak{V}) with (upsilon le N) such that the Diophantine inequality (left|{lambda }_{2}{p}_{2}^{2}+{lambda }_{3}{p}_{3}^{3}+{lambda }_{4}{p}_{4}^{4}+{lambda }_{5}{p}_{5}^{5}-upsilon right|<{upsilon }^{-delta }) has no solution in primes p2, p3, p4, p5. In this paper, we prove that for any (varepsilon >0,Eleft(mathfrak{V},N,delta right)ll {N}^{1-19/378+2delta +varepsilon },) which refines the previous result.
We show the existence of nodal solutions of the second-order nonlinear boundary value problem
$$begin{array}{l}-{u}^{^{primeprime} }left(xright)=lambda left(gleft(uleft(xright)right)+pleft(x,uleft(xright),{u}^{mathrm{^{prime}}}left(xright)right)right),xin left(mathrm{0,1}right), uleft(0right)=uleft(1right)=0,end{array} ({text{P}})$$where λ > 0 is a parameter, p : [0, 1]×ℝ2 → ℝ and g : ℝ →ℝ are continuous functions, and g(0) = 0. For a nonnegative integer k, we say that a solution is nodal if it has only simple zeros in (0, 1) and has exactly k-1 such zeros. Under some suitable conditions, we obtain that there exists λ∗ > 0 (or λ∗ > 0) such that for fixed k ∈ {1, 2,…}, problem (P) has at least one nodal solution for λ ∈ (k2π2/g∞, λ∗) (or λ ∈ (λ∗, k2π2/g∞)), where g∞ = lim|s|→∞ g(s)/s. The proof of our main results relies on the bifurcation technique.
We show that the distribution class ℒ(γ) 𝒪𝒮 is not closed under infinitely divisible distribution roots for γ > 0, that is, we provide some infinitely divisible distributions belonging to the class, whereas the corresponding Lévy distributions do not. In fact, one part of these Lévy distributions belonging to the class 𝒪ℒℒ(γ) have different properties, and the other parts even do not belong to the class 𝒪ℒ. Therefore, combining with the existing related results, we give a completely negative conclusion for the subject and Embrechts–Goldie conjecture. Then we discuss some interesting issues related to the results of this paper.
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