Stochastic homogenization of a class of monotone eigenvalue problems

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form

Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

Let U(λ, μ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition

On solutions of almost hypoelliptic equations in weighted Sobolev spaces

It is proved that if P(D) is a regular, almost hypoelliptic operator and

Asymptotic formulas for nonoscillatory solutions of conditionally oscillatory half-linear equations

We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

Existence Results for a Class of <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)- Kirchhoff Problem with a Singular Weight

The present article is concerned with the following nonlocal elliptic equation involving concave and convex terms,

$$\begin{array}{ll}- M \left(\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}{\rm d}x\right)\Big(\Delta_{p(x)}u\Big) \!&=\! \lambda \big(g(x)|u|^{q(x)-2}u\!-\!h(x)\\ &\quad |u|^{r(x)-2}u\big), \quad x\in \Omega,\\ & u = 0,\quad x\in \partial\Omega. \end{array}$$

By means of the variational approach, we prove that the above problem admits a sequence of infinitely many solutions under suitable assumptions.

     

Rough bilinear fractional integrals with variable kernels

We study the rough bilinear fractional integral

Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function φ is

The generalized Roper-Suffridge extension operator in Banach spaces (III)

Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) $ \Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f) on the domain $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } $ \Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} } defined by

Nonexistence results and estimates for some nonlinear elliptic problems

Here we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type or . We give integral estimates and nonexistence results. They depend on properties of the supersolutions of the equationsL _A ^u=0,L _B ^v=0, which suppose weak coercivity conditions. Under stronger conditions, we give pointwise estimates in case of equalities, using Harnack properties.     

Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C ¹-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.     

L ∞-solutions for some nonlinear degenerate elliptic and parabolic equations

In this paper we prove the existence of bounded solutions for equations whose prototype is:

New Kamenev-type oscillation criteria for half-linear partial differential equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.     

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

An approach via symmetrization methods to nonlinear elliptic problems with a lower order term

In this paper we consider a class of nonlinear elliptic problems of the type

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1 ⩽ <Emphasis Type="Italic">q</Emphasis> < <Emphasis Type="Italic">p</Emphasis> < ∞

We consider a new Sobolev type function space called the space with multiweighted derivatives $ W_{p,\bar \alpha }^n $ W_{p,\bar \alpha }^n , where $ \bar \alpha $ \bar \alpha = (α ₀, α ₁,…, α _n ), α _i ∈ ℝ, i = 0, 1,…, n, and $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} $ \left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|} ,

The Nonexistence of the Solutions for the Non-Newtonian Filtration Equation with Absorption

The paper proves the nonexistence of the solution for the following Cauchy problem\begin{align*}\begin{cases}u_{t} ={\rm div}\left(\left|\nabla u^{m} \right|^{p-2} \nabla u^{m} \right)-\lambda \; u^{q},&\qquad \left(x,t\right)\in S_{T} ={\mathbb{R}}^N \times \left(0,T\right), \\u\left(x,\; 0\right)=\delta \left(x\right), &\qquad x\in {\mathbb{R}}^,\end{cases}\end{align*}under some conditions on \textit{m,p,q},$\lambda$, where $\delta $ is Dirac function.     

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.