Pohozaev-type Identity Research Articles

Existence and Nonexistence Results of a Problem Involving the Pseudobilaplacian Operator

In this paper we obtain some existence and nonexistence results of a problem involving the pseudobilaplacian operator. By the sub and supersolutions techniques we prove that the problem admits a positive solution when it is subhomogeneous. In the case where it is superhomogeneous, we prove the existence of nontrivial solutions. The nonexistence result follows from a Pohozaev-type identity.

An obstruction for the mean curvature of a conformal immersion 𝑆ⁿ→ℝⁿ⁺¹

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H H of a conformal immersion S n → R n + 1 S^n\to \mathbb {R}^{n+1} satisfies ∫ ∂ X H = 0 \int \partial _X H=0 where X X is a conformal vector field on S n S^n and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on S n S^n inside the standard conformal class.

Best Sobolev constants and quasi‐linear elliptic equations with critical growth on spheres

AbstractSharp existence and nonexistence results for positive solutions of quasilinear elliptic equations with critical growth in geodesic balls on spheres are established. The arguments are based on Pohozaev type identities and asymptotic estimates for Emden–Fowler type equations. By means of spherical symmetrization and the concentration‐compactness principle existence and nonexistence results for general domains on spheres are obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The solvability of quasilinear Brezis–Nirenberg-type problems with singular weights

In this paper, we consider the existence and non-existence of non-trivial solutions to quasilinear Brezis–Nirenberg-type problems with singular weights. First, we shall obtain a compact imbedding theorem which is an extension of the classical Rellich–Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev-type identity and obtain a non-existence result. Thirdly, thanks to the generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS) c condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.

On travelling wave solutions of a generalized Davey-Stewartson system

The generalized Davey-Stewartson (GDS) equations, as derived by Babaoglu & Erbay (2004, Int. J. Non-Linear Mech., 39, 941-949), is a system of three coupled equations in (2 + 1) dimensions modelling wave propagation in an infinite elastic medium. The physical parameters (γ, m 1 , m 2 , λ and n) of the system allow one to classify the equations as elliptic-elliptic-elliptic (EEE), elliptic-elliptic-hyperbolic (EEH), elliptic-hyperbolic-hyperbolic (EHH), hyperbolic-elliptic-elliptic (HEE), hyperbolic-hyperbolic-hyperbolic (HHH) and hyperbolic-elliptic-hyperbolic (HEH) (Babaoglu et al., 2004, preprint). In this note, we only consider the EEE and HEE cases and seek travelling wave solutions to GDS systems. By deriving Pohozaev-type identities we establish some necessary conditions on the parameters for the existence of travelling waves, when solutions satisfy some integrability conditions. Using the explicit solutions given in Babaoglu & Erbay (2004) we also show that the parameter constraints must be weaker in the absence of such integrability conditions.

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohozaev-type identity.

Hardy-type inequalities and Pohozaev-type identities for a class of [formula omitted]-degenerate subelliptic operators and applications

The purpose of this paper is to consider a class of p-degenerate subelliptic operators L p constructed by generalizing Greiner's vector fields. Their fundamental solutions at the origin are established with the aid of the properties of radial functions. A Picone-type identity and a Hardy-type inequality with respect to vector fields are proved. Some Pohozaev-type identities and applications to nonlinear equations are given. Finally, a Carleman-type estimate and uniqueness of the operator L 2 are discussed.

Critical exponents for semilinear equations of mixed elliptic‐hyperbolic and degenerate types

AbstractFor semilinear Gellerstedt equations with Tricomi, Goursat or Dirichlet boundary conditions we prove Pohozaev type identities and derive non existence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations. A critical exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic hyperbolic or degenerate settings where the power is one less than the critical exponent in a relevant Sobolev imbedding. © 2002 Wiley Periodicals, Inc.

The Brézis–Nirenberg Problem on [formula omitted

In this paper we study existence and nonexistence of solutions to the Brézis–Nirenberg problem for different values of λ in geodesic spheres on S3. The picture differs considerably from the one in the Euclidean space. It is shown that large spheres containing the hemisphere have two different type of radial solutions for negative values of λ. Numerical results indicate that for λ very small the solutions have a maximum near the boundary, whereas for larger values of λ the maximum is at the origin. The techniques used are: Pohozaev type identities, concentration-compactness lemma and numerical methods.

Solutions for a quasilinear elliptic equation with critical Sobolev exponent and perturbations on ${\bf R}^N$

We consider the following quasilinear elliptic problem \begin{equation} \begin{cases} -\mbox{div } |\nabla u|^{p-2}\nabla u)+c|u|^{p-2}u=|u|^ {p^*-2}u+f(x,u)+h(x)\\ u\in W^{1, p}(R^N), \ \ \ N>p\geq 2, \end{cases} \tag*{*} \end{equation} where $c>0, $ $ p^*=\frac{Np}{N-p},$ $ h(x)\in W^{-1, \frac{p}{p-1}}(R^N)$ (i.e., the dual space of $W^{1, p}(R^N)$), $f(x,0)=0 $ and $ f(x,u)$ is a lower-order perturbation of $|u|^ {p^*-2}u$ in the sense that $\lim_{u\rightarrow \infty}\frac{f(x,u)}{|u|^ {p^*-2}u}=0$. It is well known that (*) has only a trivial solution if $f(x,u)\equiv h(x)\equiv 0$ by a Pohozaev type identity, but (*) has a nontrivial solution if there is a subcritical perturbation, e.g., $h(x){\equiv} 0$ and $f(x,u){\not\equiv} 0$. In this paper, we prove that (*) has at least two distinct solutions if there are two perturbations, i.e., $f(x,u) {\not\equiv} 0 $ and $ h(x){\not\equiv} 0$ (inhomogeneous term) with $\| h\|$ small enough.

Solitary waves for two–dimensional Schr dinger–Kadomtsev–Petviashvili equations

In this paper, the authors discuss the existence and non–existence of solitary waves for the coupled Schrodinger–Kadomtsev–Petviashvili equations in R2. The non–existence is proved by deriving the so–called Pohozaev–type identity, and the existence is obtained by considering a minimization problem using the concentration–compactness principle.

Uniqueness of positive radial solutions of $\Delta u+K(\vert x\vert )\gamma(u)=0$

We investigate the global structure of positive radial solutions of a semilinear elliptic equation $\Delta u+K(|x|)\gamma (u)=0$, and study the uniqueness of ground state solutions of this equation. Our discussion is based on a Pohozaev-type identity and some detailed investigation for the oscillatory and asymptotic behavior of the solutions and their variational functions.

Uniqueness Theorems for Positive Radial Solutions of Quasilinear Elliptic Equations in a Ball

We establish a new Pohozaev-type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div(|∇u|m−2emsp14;∇u)+f(u)=0 inB, andu=0 on ∂B, whereBis a finite ball in Rn,n⩾3 and 1<m⩽n. Applying this main uniqueness result we can prove that the semilinear problemΔu+μup+uq=0 inB, andu=0 on ∂B, whereμ>0 and 1⩽p<q⩽(n+2)/(n−2), has a unique positive solution whenn⩾6. This gives a complete answer to an open problem raised by Brezis and Nirenberg in 1983 in the casen⩾6. We shall also derive some partial results to the open problem in the casesn=3, 4, 5.