Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper, we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n$ $(n ≥1)$, where $b ≥0$, $γ ≥1$, $a_i ≥1$, $µ$, $b_i >0$ $(i =1,2)$, $D$, $S∈ C^2([0,∞))$ fulfilling $D(s) ≥ a_0(s+1)^{−α}$, $0 ≤ S(s) ≤ b_0(s+1)^β$ for all $s ≥ 0,$ where $a_0,b_0 > 0$ and $α,β ∈ \mathbb{R}$ are constants. The purpose of this paper is to prove that if $b ≥ 0$ and $µ > 0$ sufficiently large, the globally bounded solution $(u,v,w)$ with nonnegative initial data $(u_0,v_0,w_0)$ satisfies $$\Big\| u(·,t)− \Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\|_{L^∞(Ω)}+\Big\| v(·,t)−\frac{b_1b_2}{a_1a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)} +\Big\| w(·,t)−\frac{b_2}{a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)}→0$$ as $t→∞$.     

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta <n\) and \(\tau >0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have

$$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$

Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta <2m\) and \(\tau >0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that

$$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$

for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use

$$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$

The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$

where the nonlinearity f has the critical exponential growth.

     

Existence of Nontrivial Weak Solutions to Quasi-linear Elliptic Equations with Exponential Growth

In this paper, we study the existence of nontrivial weak solutions to the following quasi-linear elliptic equations $$-Δ_nu+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}, x ∈ R^n(n ≥ 2),$$ where $-Δ_nu=-div(|∇u|^{n-2}∇u), 0 ≤β < n, V:R^n→R$ is a continuous function, f (x,u) is continuous in $R^n×R$ and behaves like $e^{αu^{\frac{n}{n-1}}}$ as $u→+∞$.     

Low Mach number limit of multidimensional steady flows on the airfoil problem

Given $$\alpha >0$$, we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$where $$W^{1,n}_{0,\mathrm{rad}}(B)$$ is the usual Sobolev spaces of radially symmetric functions on B in $${\mathbb {R}}^n$$ with $$n\ge 2$$. Without restricting to the class of functions $$W^{1,n}_{0,\mathrm{rad}}(B)$$, we should emphasize that the above inequalities fail in $$W^{1,n}_{0}(B)$$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.     

Global W2,p (2<=p

This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.     

On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.     

Regularity of Positive Solutions for an Integral System on Heisenberg Group

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} for $x\in \mathbb{H}^n$, where $0<\alpha 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

Existence of C1-solutions to Certain Non-uniformly Degenerate Elliptic Equations

We are concerned with the Dirichlet problem of {div A(x, Du) + B(z) = 0 \qquad in Ω u= u_0 \qquad \qquad on ∂ Ω Here Ω ⊂ R^N is a bounded domain, A(x, p) = (A¹ (x, p), ... >A^N (x, p}) satisfies min{|p|^{1+α}, |p|^{1+β}} ≤ A(x, p) ⋅ p ≤ α_0(|p|^{1+α}+|p|^{1+β}) with 0 < α ≤ β. We show that if A is Lipschitz, B and u_0 are bounded and β < max {\frac{N+2}{N}α + \frac{2}{N},α + 2}, then there exists a C¹-weak solution of (0.1).     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

THROREMS OF DISTORTION FOR SCHLICHT FUNCTIONS

Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l} {\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m}, \end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m. Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l} \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3, \end{array}\], where \[\begin{array}{l} {F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}). \end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun. Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l} P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2}, \end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing.     

A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight

The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{ \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω,\end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$. We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.     

Impulsive fractional boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplace operator

In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem

$$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$

where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and

$$\begin{aligned} \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right) \\&- {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^-\right) \right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right)= & {} \lim _{t \rightarrow t_j^+} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j^-)\right)= & {} \lim _{t\rightarrow t_j^-}{_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) . \end{aligned}$$

By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.

     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.     

Uniform Mean Value Estimates and Discrete Hilbert Inequalities VIA Orthogonal Dirichlet Series

Let \({\{\lambda_j\}^\infty_{j=0}}\) be a strictly increasing sequence of positive numbers with λ₀ = 0 and λ₁ = 1. We use orthogonal Dirichlet polynomials associated with the arctangent density, to observe that for r > 0, $$\begin{array}{ll}\int^\infty_0\left |\sum\limits^\infty_{n=1}(-1)^{n-1}a_n\lambda^{-irt}_n\right |^2 \frac{dt}{\pi(1 + t^2)}\\ = \sum\limits^\infty_{n=1}(\lambda^{2r}_n - \lambda^{2r}_{n-1})\left |\sum\limits^\infty_{k=n}(-1)^{k-1}\frac{ak}{\lambda^r_k}\right |^2,\end{array}$$ when the right-hand side converges. As a consequence, we obtain uniform mean value estimates, discrete Hilbert type inequalities, and asymptotics as r → ∞ for classes of Dirichlet series.     

Supercritical Elliptic Equation in Hyperbolic Space

In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.     

On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators

In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.     

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn

We establish sufficient conditions under which the quasilinear equation $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.