Pohozaev Type Research Articles

A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space

We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic spaceΔBNu+λu+|u|p−1u+θuln⁡u2=0,u∈H1(BN),u>0inBN, and study the existence vs non-existence results. We show that whenever θ>0, there exists an H1-solution, while for θ<0, there does not exist a positive solution in a reasonably general class. Since the perturbation uln⁡u2 changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an “almost” precise lower asymptotic decay estimate on the positive solutions for θ<0, culminating in proving their non-existence assertion.

Asymptotic Analysis for Sacks-Uhlenbeck 𝛼-Harmonic Maps From Degenerating Riemann Surfaces

In this paper, we investigate the asymptotic analysis and qualitative behaviour for a general sequence of Sacks-Uhlenbeck α \alpha -harmonic maps from degenerating Riemann surfaces. This answers an open problem proposed by J. D. Moore, aiming at developing a partial Morse theory for closed parametrized minimal surfaces with arbitrary codimensions in compact Riemannian manifolds. In technical terms, we shall deal with three types of necks which give rise to new difficulties. We establish a generalized energy identity involving some new quantities, such as Pohozaev type constants which measure the extent to which the Pohozaev type identity fails, and some locating constants which characterize the limit position of the blow-up points in long hyperbolic cylinders. Furthermore, we show that the limit of necks are all geodesics in the target manifold where some length formulas are given. Finally, we exploit some geometric and topological conditions to ensure that the limiting neck geodesics appeared in a blow-up sequence of α \alpha -harmonic maps with finite Morse index are all of finite length, which implies that energy identity hold.

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.

Existence of positive ground state solutions of Nehari–Pohozaev type for a subcritical and critical coupled system of Kirchhoff–Schrödinger equations in $ \mathbb{R}^3 $

Existence of positive ground state solutions of Nehari–Pohozaev type for a subcritical and critical coupled system of Kirchhoff–Schrödinger equations in $ \mathbb{R}^3 $

A Study of Biharmonic Equation Involving Nonlocal Terms and Critical Sobolev Exponent

AbstractIn this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of the equation $$\begin{aligned} \Delta ^{2} u= \Big (|x|^{-\theta }*|u|^{p_{\theta }}\Big )|u|^{p_{\theta }-2}u +\alpha \Big (|x|^{-\gamma }*|u|^{p}\Big )|u|^{p-2}u \quad \text{ in } {{\mathbb {R}}}^{N}, \end{aligned}$$ Δ 2 u = ( | x | - θ ∗ | u | p θ ) | u | p θ - 2 u + α ( | x | - γ ∗ | u | p ) | u | p - 2 u in R N , where $$0&lt;p\le p_{\gamma }^{*}$$ 0 &lt; p ≤ p γ ∗ , $$\alpha &gt;0$$ α &gt; 0 , $$\theta , \gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$p_{\theta }=\frac{2(N-\theta )}{N-4}$$ p θ = 2 ( N - θ ) N - 4 and $$N\ge 5$$ N ≥ 5 . Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.

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.

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

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.

Existence of solutions for a class of quasilinear elliptic equations involving the p-Laplacian

This paper is concerned with the existence of solutions for the quasilinear elliptic equations − Δ p u − Δ p ( | u | 2 α ) | u | 2 α − 2 u + V ( x ) | u | p − 2 u = | u | q − 2 u , x ∈ R N , where α ≥ 1 , 1<p<N, p ∗ = Np / ( N − p ) , Δ p is the p-Laplace operator and the potential 0 $ ]]> V ( x ) > 0 is a continuous function. In this work, we mainly focus on nontrivial solutions. When 2 αp < q < p ∗ , we establish the existence of nontrivial solutions by using Mountain-Pass lemma; when q ≥ 2 α p ∗ , by using a Pohozaev type variational identity, we prove that the equation has no nontrivial solutions.

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.

Harmonic Maps with Free Boundary from Degenerating Bordered Riemann Surfaces

We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.

Global weak solutions to the stochastic Ericksen–Leslie system in dimension two

&lt;p style='text-indent:20px;'&gt;We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.&lt;/p&gt;

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.

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.

Solutions to a fourth-order elliptic equation with steep potential

In this paper, we consider the following fourth-order elliptic equation with steep potential and concave–convex nonlinearities Δ2u−(1+a∫R3|∇u|2dx)Δu+Vλ(x)u=f(x)|u|q−2u+g(x)|u|p−2u,x∈R3, existence and multiplicity of solutions to above equation are proved by using variational method. The usual Nehari manifold and Pohozaev type manifold do not work here. Our results can be viewed as an extension of the existing results.

A Berestycki-Lions' type result to a quasilinear elliptic problem involving the 1-Laplacian operator

In this work we study a quasilinear elliptic problem involving the 1-Laplacian operator in RN, whose nonlinearity satisfy conditions similar to those ones of the classical work of Berestycki and Lions. Several difficulties are faced when trying to generalize the arguments of the semilinear case, to this quasilinear problem. The main existence theorem is proved through a new version of the well known Mountain Pass Theorem to locally Lipschitz functionals, where it is considered the Cerami compactness condition rather than the Palais-Smale one. It is also proved that all bounded variation solutions which are regular enough, satisfy a Pohozaev type identity.

Positive solution to Schrödinger equation with singular potential and double critical exponents

In this paper, we consider the following Schrödinger equation with singular potential and double critical exponents: -\Delta u + \frac{A}{|x|^{\alpha}}u = |u|^{2^{*}-2}u + |u|^{p-2}u + \lambda |u|^{2_{\alpha}^{*}-2}u,\quad x\in \mathbb{R}^{N}, where N\geq 3 , \alpha\in(0,2) , p\in(\frac{2N-4+2\alpha}{N-2},2^{*}) , and A,\lambda&gt;0 are two real constants, and 2^{*}=\frac{2N}{N-2} is the Sobolev critical exponent, and 2_{\alpha}^{*}=2+\frac{4\alpha}{2N-2-\alpha} is the critical exponent with respect to the parameter \alpha . First, using the refined Sobolev inequality, we establish the Lions type theorem. Second, we prove that any nonnegative weak solutions of above equation satisfy Pohozaev type identity. Finally, by using perturbation method, Pohozaev type identity and Lions type theorem, we show the existence of positive solution to above equation. We point out that the double critical exponents is an new phenomenon, and we are the first to consider it. Our result partial extends the results in Badiale and Rolando [Rend. Lincei Mat. Appl. 17 (2006)], and Su, Wang and Willem [Commun. Contemp. Math. 9 (2007)].