Multiple positive solutions for a nonlinear Schr?dinger equation

We are interested in positive entire solutions of the nonlinear Schrödinger equation -Du+(la(x)+1)u = u^p-\Delta u+(\lambda a(x)+1)u = u^p where a ≤ 0 has a potential well and p > 1 is subcritical. Using variational methods we prove the existence of multiple positive solutions which localize near the potential well int(a^-1(0)) for l\lambda large.     

Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation

Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem

Solvability of boundary-value problems for nonlinear fractional differential equations

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

Harmonicity of functions satisfying a weak form of the mean value property

Let W ì \Bbb Rⁿ\Omega \subset {\Bbb R}^n be a smooth domain and let u ? C⁰(W).u \in C^0(\Omega ). A classical result of potential theory states that¶¶-ò_{Sr([`(x)])</sub> u(x)ds(x)=u([`(x)])-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)¶¶for every [`(x)] ? W\bar x\in \Omega and r > 0r>0 if and only if¶¶Du=0 in W.\Delta u=0 \hbox { in } \Omega.¶¶Here -ò<sub>Sr([`(x)])} u(x)ds(x)-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x) denotes the average of u on the sphere S_r([`(x)])S_r(\bar x) of center [`(x)]\bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W^2,1(W)u\in W^{2,1}(\Omega ) and let x ? Wx\in \Omega be a Lebesgue point of Du\Delta u such that¶¶-ò_Sr([`(x)]) u d s- a = o(r²)-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)¶¶for some a ? \Bbb R\alpha \in \Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.\Delta u(x)=0.     

Best constant in a three-parameter Poincaré inequality

We study the problem of finding the best constant in the generalized Poincaré inequality

Asymptotically autonomous parabolic evolution equations

We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \] If A(t) A(t) converges to a sectorial operator A with s(A)?i \Bbb R = ? \sigma(A)\cap i \Bbb R =\emptyset as t?￥ t\to\infty , then the evolution family solving the homogeneous problem has exponential dichotomy. If also f(t)? f_￥ f(t)\to f_\infty , then the solution u converges to the 'stationary solution at infinity', i.e., lim_t?￥u(t) = -A\sp-1f_￥=:u_￥,        lim_t?￥u￠(t)=0,        lim_t?￥A(t)u(t)=Au_￥. \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. .     

Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space (H,á·,·?){(\mathcal H,\langle\cdot,\cdot\rangle)}, and interval L ì (0,+￥){\Lambda\subset(0,+\infty)} and a map K ? C²(H,\mathbb R){K\in C^2(\mathcal H,\mathbb R)} whose gradient is a compact mapping, we consider the family of functionals of the type:

A two-point problem for first-order systems

Let f be a continuous function from [a, b] ×\mathbbRⁿ [a, b] \times \mathbb{R}^n into \mathbbRⁿ \mathbb{R}^n . In this paper we prove that the problem¶¶ { llu^￠ = f(t,u)+ lu(a)=u(b)=0  \left \{ \begin{array}{ll}u^{\prime}= f(t,u)+ \lambda \\[3pt]u(a)=u(b)=0\end{array}\right.\ ¶¶ has a (classical) solution for a wide class of functions f. Next we point out a particular case.     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

Nonexistence of single blow-up solutions for a nonlinear Schr?dinger equation involving critical Sobolev exponent

In this paper we shall consider the critical elliptic equation $ -\triangle u + \lambda a(x) u = u^{(N+2)/(N-2)}, \ \ x\in \Bbb R^N, \\ u > 0, \quad \int_{\Bbb R^N} |\nabla u|^2 \, dx < + \infty, \quad\quad (0.1)$ -\triangle u + \lambda a(x) u = u^{(N+2)/(N-2)}, \ \ x\in \Bbb R^N, \\ u > 0, \quad \int_{\Bbb R^N} |\nabla u|^2 \, dx < + \infty, \quad\quad (0.1) where $\lambda >0, N > 4$\lambda >0, N > 4 and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions u_lu_\lambda which blows-up and concentrates as l? +￥\lambda \to +\infty at some zero point x₀ of a(x) if the order of flatness of the function a(x) at x₀ is b ? [2,N-4)\beta\in[2,N-4)     

Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations

Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

Numerical Computation of First-Passage Times of Increasing Lévy Processes

Let {D(s), s ≥ 0} be a non-decreasing Lévy process. The first-hitting time process {E(t), t ≥ 0} (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$E(t) = \inf \{s: D(s) > t \} is a process which has arisen in many applications. Of particular interest is the mean first-hitting time U(t)=\mathbbEE(t)U(t)=\mathbb{E}E(t). This function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function [(U)\tilde](l) = (lf(l))^-1\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}, where ϕ is the Lévy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.     

Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

Girsanov theorem for stochastic flows with interaction

We prove an analog of the Girsanov theorem for the stochastic differential equations with interaction

On the time continuity of entropy solutions

We show that any entropy solution u of a convection diffusion equation ?_t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L¹_loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution.     

Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

We consider the Neumann initial boundary-value problem for the equation