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 the Sobolev embedding theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration–Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.     

Multiple positive solutions for a semilinear elliptic equation with critical Sobolev exponent

In this paper, we study a class of semilinear elliptic equations with Hardy potential and critical Sobolev exponent. By means of the Ekeland variational principle and Mountain Pass theorem, multiple positive solutions are obtained.     

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

Positive solution to semilinear parabolic equation associated with critical Sobolev exponent

We study the semilinear parabolic equation ${u_{t}- \Delta u = u^{p}, u \geq 0}$ on the whole space R ^N , ${N \geq 3}$ associated with the critical Sobolev exponent p = (N + 2)/(N ? 2). Similarly to the bounded domain case, there is threshold blowup modulus concerning the blowup in finite time. Furthermore, global in time behavior of the threshold solution is prescribed in connection with the energy level, blowup rate, and symmetry.     

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 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 $\text{∂Ω}$.     

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.     

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator Iα there are proved weighted estimates

     

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.     

Existence of solitary wave solutions for the nonlinear Klein-Gordon equation coupled with Born-Infeld theory with critical Sobolev exponent

In this paper, we prove the existence of solutions for the nonlinear Klein-Gordon equation coupled with Born-Infeld theory under the electrostatic solitary wave ansatz by using variational methods.     

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p _* : =  p(N  −  1)/(N  −  p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.      

Multiple solutions of an inhomogeneous Neumann problem for an elliptic system with critical Sobolev exponent

In this paper we prove the existence of two solutions for the inhomogeneous Neumann problem with critical Sobolev exponent.     

On an obstacle problem for degenerate elliptic operators involving the critical Sobolev exponent

We establish the existence of a solution to the variational inequality (the obstacle problem) (1.1) which involves the critical Sobolev exponent. This result is also extended to an obstacle problem with a lower order perturbation. Dedicated to Professor F. Browder on the occasion of his 80-th birthday     

A uniqueness result for a Neumann problem involving the critical Sobolev exponent

 In this paper we consider the problem

Existence of solutions for the p-Laplacian with critical Sobolev exponent and convection

In this article, we study the existence of solutions for the p-Laplacian involving critical Sobolev exponent and convection based on the theory of the Leray–Schauder degree for non-compact mappings.