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1.
In this paper, we mainly consider the stability of blowup of solutions for the p-Laplace equation with nonlinear source ${u_t = {div}(|\nabla u|^{p-2}\nabla u) + u^q,\;\;(x,t)\in\mathbb{R}^N \times (0,T)}$ , with the initial value ${u(x,0) = u_0(x) \geq 0}$ , where ${\|u_0 (x)\|_{L^\infty} \leq M}$ and T < ∞ is the blowup time. Under a small oscillation around the radial initial value, we can prove the solution blows up in finite time and obtain the blowup rate estimate of the form ${\|u(\cdot,t)\|_{L^\infty}\leq C(T-t)^{-\frac{1}{q-1}}}$ , where the constant C > 0 is dependent only on N, p, q, and the parameters q and p are expected to be ${p > 2, p-1 < q < \frac{Np}{(N-p)}_+ -1}$ .  相似文献   

2.
In this paper we consider critical points of the following nonlocal energy $$\begin{array}{ll}{\mathcal{L}}_n(u) = \int_{{I\!\!R}^n}| ({-\Delta})^{n/4} u(x)|^2 dx, \qquad(1)\end{array}$$ where ${u \in \dot{H}^{n/2}({I\!\!R}^n,{\mathcal{N}}), {\mathcal{N}} \subset {I\!\!R}^m}$ is a compact k dimensional smooth manifold without boundary and n > 1 is an odd integer. Such critical points are called n/2-harmonic maps into ${{\mathcal{N}}}$ . We prove that ${(-\Delta) ^{n/4} u\in L^p_{loc}({I\!\!R}^n)}$ for every p ≥  1 and thus ${u \in C^{0,\alpha}_{loc}({I\!\!R}^n)}$ , for every 0 < α < 1. The local Hölder continuity of n/2-harmonic maps is based on regularity results obtained in [4] for nonlocal Schrödinger systems with an antisymmetric potential and on some new 3-terms commutators estimates.  相似文献   

3.
In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where ${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$ is a continuous function, ${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$ are fixed, m ≥ 3 for pk (m ≥ 4 for p > k), ${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$ . We give an example to demonstrate our results.  相似文献   

4.
For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .  相似文献   

5.
Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let ${\mathcal{S} = \{T(s): 0 \leq s < \infty\}}$ be a nonexpansive semigroup on E such that ${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$ , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and ${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$ . When the sequences of real numbers {α n } and {t n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to ${\tilde{x}}$ , where ${\tilde{x}}$ is the unique solution in ${Fix(\mathcal{S})}$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.  相似文献   

6.
In this paper, firstly, we investigate a class of singular eigenvalue problems with the perturbed Hardy–Sobolev operator, and obtain some properties of the eigenvalues and the eigenfunctions. (i.e. existence, simplicity, isolation and comparison results). Secondly, applying these properties of eigenvalue problem, and the linking theorem for two symmetric cones in Banach space, we discuss the following singular elliptic problem $$\left\{\begin{array}{ll}-\Delta_{p}u-a(x)\frac{|u|^{p-2}u}{|x|^{p}}= \lambda \eta(x)|u|^{p-2}u+ f(x,u) \quad x \in \Omega, \\ u =0 \quad\quad\quad\quad\quad\quad\quad x\in\partial \Omega, \end{array} \right.$$ where ${a(x)=(\frac{n-p}{p})^{p}q(x),}$ if 1 < p < n, ${a(x)=(\frac{n-1}{n})^{n} \frac{q(x)}{({\rm log}\frac{R}{|x|})^{n}},}$ if p = n, and prove the existence of a nontrivial weak solution for any ${\lambda \in \mathbb{R}.}$   相似文献   

7.
We study the following nonlinear Schrödinger equations $$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$ for ${w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}$ , where g(|w|)w is super linear and subcritical, 2* = 2N/(N ? 2) if N > 2 and =  if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min Vτ 0 for some τ0 > 0, or ${\max W > \kappa_{0}}$ for some ${\kappa_0 > 0}$ .  相似文献   

8.
In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.  相似文献   

9.
We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure μ with even continuous density and sections are of arbitrary dimension n ? k, 1 ≤ kn. If K is a generalized k-intersection body, then $$\mu(K)\,\leq\,\frac{n}{n-k}c_{n,k} \max_{H} \mu(K\cap H) {\rm Vol}_n(K)^{k/n}.$$ Here ${c_{n,k} = |B_2^n|^{(n-k)/n}/|B_2^{n-k}| <1 ,{ }|B_2^n|}$ is the volume of the unit Euclidean ball, and maximum is taken over all (n ? k)-dimensional subspaces of ${\mathbb{R}^{n}}$ . The constant is optimal, and for each intersection body the inequality holds for every k. We also prove a stronger “difference” inequality. The proof is based on stability in the lower dimensional Busemann–Petty problem for arbitrary measures in the following sense. Let ${\varepsilon >0 ,\ 1\le k < n}$ . Suppose that K and L are origin-symmetric star bodies in ${\mathbb{R}^{n}}$ , and K is a generalized k-intersection body. If for every (n ? k)-dimensional subspace H of ${\mathbb{R}^{n}}$ $$\mu(K\cap H)\leq \mu(L\cap H)+\varepsilon,$$ then $$\mu(K)\leq \mu(L) +\frac{n}{n-k}c_{n,k} {\rm Vol}_n(K)^{k/n} \varepsilon.$$   相似文献   

10.
Let ${\mathcal L(r) = \sum_{n=0}^\infty a_nr^{\lambda_n}}$ be a lacunary series converging for 0 <  r < 1, with coefficients in a quasinormed space. It is proved that $$\int_0^1 F(1-r,\|\mathcal L(r)\|)(1-r)^{-1}\,{\rm d}r < \infty $$ if and only if $$ \sum_{n=0}^\infty F(1/\lambda_n,\|a_n\|) < \infty, $$ where F is a “normal function” of two variables. In the case when p ≥ 1 and F(x, y) =  x y p , this reduces to a theorem of Gurariy and Matsaev. As an application we prove that if ${f(r\zeta) = \sum_{n=0}^\infty r^{\lambda_n}f_{\lambda_n}(\zeta)}$ is a function harmonic in the unit ball of ${\mathbb R^N,}$ then $$\int_0^1M_p^q(r,f)(1-r)^{q\alpha-1} \,{\rm d}r <\infty\quad (p,\,q,\,\alpha >0 ) $$ if and only if $$\sum_{n=0}^\infty \|f_{\lambda_n} \|^q_{L^p(\partial B_N)}(1/\lambda_n)^{q\alpha} <\infty. $$   相似文献   

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