Pohozaev-type Identity Research Articles

Nonexistence of multi-dimensional solitary waves for the Euler–Poisson system

We study the nonexistence of multi-dimensional solitary waves for the Euler–Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler–Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves in the L1 space for any traveling velocity and for general pressure laws. Our results provide theoretical evidence for the stability of line solitary waves in multi-dimensional Euler–Poisson flows. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler–Poisson system for ions and electrons.

Existence and nonexistence of solutions for the mean curvature equation with weights

In this paper we study existence and nonexistence of positive radial solutions of a Dirichlet problem for the prescribed mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem under consideration appears rather delicate, it requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity. In addition, sufficient conditions for global solutions to be oscillatory are given.

On the Choquard equation with double critical Sobolev exponents in ℝ N

In this paper, we study the existence and nonexistence results to the Choquard equation − Δ u = ( ∫ R N | u ( y ) | 2 α ∗ | x − y | α d y ) | u | 2 α ∗ − 2 u ± | u | q − 2 u in R N , where 2 α ∗ = 2 N − α N − 2 , 0 < α < N , 1 < q ≤ 2 ∗ , 2 ∗ = 2 N N − 2 , N ≥ 3 . We first use the Pohozaev-type identity to show the nonexistence of solutions for 1 < q < 2 ∗ . When the equation has double critical exponents, i.e. q = 2 ∗ , we obtain the existence of radial ground state solutions by the Nehari manifold and Mountain pass theorem.

Blow-up solutions for a 4-dimensional semilinear elliptic system of Liouville type in some general cases

This paper is devoted to the existence of singular limit solutions for a nonlinear elliptic system of Liouville type under Navier boundary conditions in a bounded open domain of R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{4}$\\end{document}. The concerned results are obtained employing the nonlinear domain decomposition method and a Pohozaev-type identity.

Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$

We show that axially symmetric solutions on \mathbb{S}^4 to a constant Q -curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter \alpha in front of the Paneitz operator belongs to the interval [\frac{473 + \sqrt{209329}}{1800}\approx 0.517, 1) . This is in contrast to the case \alpha=1 , where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on \mathbb{S}^2 . As a consequence, we prove an improved Beckner's inequality on \mathbb{S}^4 for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when \alpha=1/5 by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for \alpha \in (1/5, 1/2) via a bifurcation method.

Regularity and Pohozaev identity for the Choquard equation involving the [formula omitted]-Laplacian operator

In this paper, we study the regularity of weak solutions for a class of nonlinear Choquard equations driven by the p-Laplacian operator. We also establish a Pohozaev type identity.

Singular limit solutions for a four‐dimensional semilinear elliptic Kuramoto–Sivashinsky system in some general case

We consider the existence of singular limit solutions for a nonlinear elliptic Kuramoto–Sivashinsky system with Navier boundary conditions. We use the nonlinear domain decomposition method and a Pohozaev type identity.

A generalized fractional Pohozaev identity and applications

Abstract We prove a fractional Pohozaev-type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to the nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.

Ergodic Mean-Field Games with aggregation of Choquard-type

We consider second-order ergodic Mean-Field Games systems in the whole space RN with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be +∞.

Regular solutions to elliptic equations

A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is presented. Another appendix on the phase plane analysis as needed to take advantage of initial conditions is also included. For studies on singular solutions the reader is referred to Ardila et al., Milan J. Math (2014) and references therein. See also https://ejde.math.txstate.edu/special/02/c2/abstr.html

Sobolev-type regularity and Pohozaev-type identities for some degenerate and singular problems

Sobolev-type regularity results are proved for solutions to a class of second-order elliptic equations with a singular or degenerate weight, under non-homogeneous Neumann conditions. As an application a Pohozaev-type identity for weak solutions is derived.

Existence of solutions to a class of quasilinear elliptic equations

This paper is concerned with the existence and nonexistence of nontrivial solutions for a class of quasilinear elliptic equations in RN involving the p-Laplacian. By utilizing a change of variables, the quasilinear equations are reduced to semilinear equations, whose associated functionals satisfy the geometric hypotheses of the mountain pass theorem. We establish the existence of nontrivial solutions via Mountain-Pass theorem. Furthermore, by using a Pohozaev type identity we prove the nonexistence of nontrivial solution under certain conditions.

An exterior overdetermined problem for Finsler N-Laplacian in convex cones

We consider a partially overdetermined problem for anisotropic N-Laplace equations in a convex cone Sigma intersected with the exterior of a bounded domain Omega in {mathbb {R}}^N, Nge 2. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that Sigma cap Omega must be the intersection of the Wulff shape and Sigma . Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.

Planar Schrödinger-Poisson system with critical exponential growth in the zero mass case

In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero mass{−Δu+ϕu=f(u),x∈R2,Δϕ=2πu2,x∈R2, where f∈C(R,R) has the critical exponential growth at infinity and there is no monotonicity restriction on f(u)/u3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ=liminf|t|→∞t2F(t)eα0t2>0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t2F(t)eα0t2 or tf(t)eα0t2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) [21] dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V(x)u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: −Δu=f(u) in R2.

Uniqueness and Liouville type results for radial solutions of some classes of k -Hessian equations

We establish a uniqueness theorem and a Liouville type result for positive radial solutions of some classes of nonlinear autonomous equation with the k -Hessian operator. We also give some interesting qualitative properties of solutions. We provide an approach, based upon a Pohozaev-type identity, that unifies all our results.

On the Rosenau equation: Lie symmetries, periodic solutions and solitary wave dynamics

In this paper, we first consider the Rosenau equation with the quadratic nonlinearity and identify its Lie symmetry algebra. We obtain reductions of the equation to ODEs, and find periodic analytical solutions in terms of elliptic functions. Then, considering a general power-type nonlinearity, we prove the non-existence of solitary waves for some parameters using Pohozaev type identities. The Fourier pseudo-spectral method is proposed for the Rosenau equation with this single power type nonlinearity. In order to investigate the solitary wave dynamics, we generate the initial solitary wave profile by using the Petviashvili’s method. Then the evolution of the single solitary wave and overtaking collision of solitary waves are investigated by various numerical experiments.

On the fractional relativistic Schrödinger operator

We collect some interesting results for equations driven by the fractional relativistic Schrödinger operator (−Δ+m2)s with s∈(0,1) and m>0. More precisely, for the linear theory, we prove Hölder-Schauder-Zygmund regularity results and a Kato's inequality. For the nonlinear theory, we obtain L∞-regularity, exponential decay, a Pohozaev-type identity, and a symmetry result for solutions of certain nonlinear fractional problems.

Ground states for fractional linear coupled systems via profile decomposition * *The first and second authors were partially supported by Capes/Brazil. The third author was partially supported by CNPq under Grant 429955/2018-9.

In this work we study existence and nonexistence of weak and ground state (least energy) solutions for a class of nonlocal linearly coupled elliptic systems. We deal with nonautonomous nonlinearities that may not satisfy any kind of monotonicity, also the related potentials may not have any kind of smoothness. In order to obtain ground states, instead of applying the well known methods of Nehari–Pohozaev manifold, we introduce new arguments and techniques whose are based on a Pohozaev type identity, a concentration-compactness principle and a profile decomposition type result.

Positive solutions for second-order differential equations o Kirchhoff type on the half-line

The aim of the present paper is to study the existence of nontrivial nonnegative solutions for a second-order boundary value problem of Kirchhoff type on the half-line. Our approach is based on variational methods, a monotonicity trick related to the mountain pass lemma, cut-off functional technique, and a Pohozaev type identity.

Pohozaev-type identities for a pseudo-relativistic Schrödinger operator and applications

We prove a Pohozaev-type identity for both the problem in and its harmonic extension to when 0<s<1. So, our setting includes the pseudo-relativistic operator and the results showed here are original, to the best of our knowledge. The identity is first obtained in the extension setting and then ‘translated’ into the original problem. In order to do that, we develop a specific Fourier transform theory for the fractionary operator , which lead us to define a weak solution u to the original problem if the identity (S) is satisfied by all . The obtained Pohozaev-type identity is then applied to prove both a result of non-existence of solution to the case if and a result of existence of a ground state, if f is modeled by , for a constant κ. In this last case, we apply the Nehari–Pohozaev manifold introduced by D. Ruiz. Finally, we inform that positive solutions of are radially symmetric and decreasing with respect to the origin, if f is modeled by functions like , or .