On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent

In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponent$\text{Δ}{}^2\text{u=u}{}^{\mathrm{n+4/(n-4)}}\text{in}\text{Ω,}\text{u>0}\text{in}\text{Ω,}\text{u=}\text{Δ}\text{u=0}\text{on}\text{∂Ω,}$where $\text{Ω}$ is a bounded domain of $\text{R}{}^{\mathrm{n}}\text{(n⩾5)}$ with a smooth boundary $\text{∂Ω}$.     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

On the critical nonlinear damped wave equation with large initial data

We study the nonlinear damped wave equation

(0.1)     

On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P _−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x ₀ ∈Ω as ε → 0, moreover x ₀ is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x ₀ of the Robin’s function, there exist solutions of (P _−ε) concentrating around x ₀ as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P _−ε), the supercritical problem (P _+ε) has no solutions which concentrate around a point of Ω as ε → 0. Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena Pacella for their constant support.     

The reaction-diffusion equation with Lewis function and critical Sobolev exponent

In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of reaction-diffusion equation with Lewis function and critical Sobolev exponent.     

On a perturbed anisotropic equation with a critical exponent

Abstract We study a perturbed anisotropic equation without using the knowledge of the limiting problem. This provides a different method from that introduced by Brézis and Nirenberg [4]. Our arguments use some tools recently developed in [5, 6]. Keywords: Anisotropic critical exponent, Critical level, Compactness, Nehari manifold Mathematics Subject Classification (2000): 35J25, 35J60, 35J65     

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

On an initial inverse problem in nonlinear heat equation associated with time-dependent coefficient

In this paper, a nonlinear backward heat problem with time-dependent coefficient in the unbounded domain is investigated. A modified regularization method is established to solve it. New error estimates for the regularized solution are given under some assumptions on the exact solution.     

A damped hyerbolic equation with critical exponent

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X^s _θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L^p (L^q ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L^p → L^q Carleman estimates, derived using the X^s _θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.     

On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent

This paper is concerned with the existence of the nontrivial solutions of the following problem:

     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .

     

On the Fujita exponent for a semilinear heat equation with a potential term

We consider the existence and nonexistence of positive global solutions for the Cauchy problem,

     

On solvability of a class of nonlinear elliptic type equation with variable exponent

In this paper, we study the Dirichlet problem for the implicit degenerate nonlinear elliptic equation with variable exponent in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions for the existence of a solution without regularization and any restriction between the exponents. Furthermore, we define the domain of the operator generated by posed problem and investigate its some properties and also its relations with known spaces that enable us to prove existence theorem.     

A critical exponent in a degenerate parabolic equation

We consider positive solutions of the Cauchy problem in $\mathbb{R\,}^n$ for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd.