Local Pohozaev Identities Research Articles

Nondegeneracy of the solutions for elliptic problem with critical exponent

This paper deals with the following nonlinear elliptic equation: −Δu=Q(|y′|,y″)uN+2N−2,u>0,inRN,u∈D1,2(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\Delta u=Q(|y'|,y'')u^{\\frac{N+2}{N-2}},\\,\\,u>0,\\,\\,\ ext{in}\\,{ \\mathbb{R}}^{N},\\,\\,u\\in D^{1,2}({\\mathbb{R}}^{N}), $$\\end{document} where (y′,y″)∈R2×RN−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(y',y'')\\in {\\mathbb{R}}^{2}\ imes {\\mathbb{R}}^{N-2}$\\end{document}, N≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N\\geq 5$\\end{document}, Q(|y′|,y″)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q(|y'|,y'')$\\end{document} is a bounded nonnegative function in R2×RN−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{2}\ imes {\\mathbb{R}}^{N-2}$\\end{document}. By using the local Pohozaev identities we prove a nondegeneracy result for the positive solutions constructed in (Peng et al. in J. Differ. Equ. 267:2503–2530, 2019).

Uniqueness and non-degeneracy of solutions for nonlinear fractional Schrödinger equation with perturbation

This paper concerned a fractional Schrödinger equation in whole space, for ɛ > 0 is a small parameter, ɛ2s(−Δ)su + V(x)u = |u|p−2u, where 12<s<1, N > 1 and 2<p<2NN−2s. We prove the non-degeneracy and uniqueness of bubble solutions by using local Pohozaev identity and finite dimensional reduction, which are the cornerstones to construct different type solutions.

Qualitative analysis to an eigenvalue problem of the Hénon equation

In this paper we study the following eigenvalue problem{−Δv=λC(α)(pα−ε)|x|αuεpα−ε−1vinΩ,u=0on∂Ω, where Ω⊂RN is a smooth bounded domain containing the origin, C(α)=(N+α)(N−2), N≥3, pα=N+2+2αN−2, α>0, ε>0 is a small parameter and uε is a single peaked solution of Hénon equation{−Δu=C(α)|x|αupα−εinΩ,u>0inΩ,u=0on∂Ω, which established by Gladiali and Grossi (2012) [21]. By using various local Pohozaev identities and blow-up analysis, we prove some asymptotic behavior of the eigenvalues λε,i and corresponding eigenfunctionsvε,i,i=2,⋯,∑1≤k<2+α2(N+2k−2)(N+k−3)!(N−2)!k!+2 when α is not an even integer. As a consequence, if 0<α<2, we have that the Morse index of the single peaked solutions is N+1, which gives an affirmative answer to a conjecture raised by Gladiali and Grossi.

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎){−Δu=up−εinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN with N≥3, p=N+2N−2 and ε>0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.In general, the solutions of (⁎) may blow-up at multiple points a1,⋯,ak of Ω as ε→0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k=1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).

Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain

In this paper we are interested in the following critical Hartree equation -Δu=(∫Ωu2μ∗(ξ)|x-ξ|μdξ)u2μ∗-1+εu,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\Delta u =\\displaystyle {\\Big (\\int _{\\Omega }\\frac{u^{2_{\\mu }^*} (\\xi )}{|x-\\xi |^{\\mu }}d\\xi \\Big )u^{2_{\\mu }^*-1}}+\\varepsilon u,~~~\ ext {in}~\\Omega ,\\\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\ ext {on}~\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where Nge 4, 0<mu le 4, varepsilon >0 is a small parameter, Omega is a bounded domain in mathbb {R}^N, and 2_{mu }^*=frac{2N-mu }{N-2} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for varepsilon small.

Asymptotic non-degeneracy of the multipeak solution for the supercritical Hénon equation

We consider the following Hénon equation with supercritical exponent: −Δu=K(|y|)up+ɛ,u>0, in B1(0),u=0, on ∂B1(0),where B1(0) is the unit ball in RN(N≥4), p=2∗−1,2∗=2NN−2, K(r)∈C1[0,1]. We prove the asymptotic non-degeneracy of multipeak solution constructed in Liu and Peng (2016) via the local Pohozaev identities.

Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem

Abstract In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , \left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V(y)u+\Phi (y)u={| u| }^{p-1}u,&amp; y\in {{\mathbb{R}}}^{3},\\ -\Delta \Phi (y)={u}^{2},&amp; y\in {{\mathbb{R}}}^{3},\end{array}\right. where ε &gt; 0 \varepsilon \gt 0 is a small parameter, 1 &lt; p &lt; 5 1\lt p\lt 5 , and V ( y ) V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.

Existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schr\"odinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schr\"odinger-Newton system. Precisely, we give the precise description of the chemical potential $\mu$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schr\"odinger-Newton system, which is markedly different from the corresponding Schr\"odinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schr\"odinger-Newton system.

Local uniqueness of concentrated solutions and some applications on nonlinear Schrödinger equations with very degenerate potentials

We revisit the following nonlinear Schrödinger equation−ε2Δu+V(x)u=up−1,u>0,u∈H1(RN), where ε>0 is a small parameter, N≥2 and 2<p<2⁎. We obtain a more accurate location for the concentrated points, the existence and the local uniqueness for positive k-peak solutions when V(x) possesses non-isolated critical points by using the modified finite dimensional reduction method based on local Pohozaev identities. Moreover, for several special potentials, with its critical point set being a low-dimensional ellipsoid, or a part of hyperboloid of one sheet or two sheets, we obtain the number and symmetry of k-peak solutions by using local uniqueness of concentrated solutions. Here the main difficulty comes from the different degenerate rate along different directions at the critical points of V(x).

Local uniqueness of multi-peak solutions to a class of Schrödinger equations with competing potential

In this paper, we consider the nonlinear Schrödinger equations. Let A(x)≔[V(x)]pp−2−N2[K(x)]−2p−2. Under some conditions on A, we show the local uniqueness of positive multi-peak solutions concentrating near k(k ≥ 2) distinct non-degenerate critical points of A by using the local Pohozaev identity. We generalize Cao–Li–Luo’s results to the competing potential cases and show how these two potentials impact the uniqueness of concentrated solutions.

Excited states for two-component Bose-Einstein condensates in dimension two

We consider the following two-component Bose-Einstein condensates (BEC) by investigating the associated L2-critical problem{−Δu1+V1(x)u1=a1u13+βu1u22+μu1inR2,−Δu2+V2(x)u2=a2u23+βu12u2+μu2inR2, with the constraint ∫R2(u12+u22)dx=1.Ground states and excited states are typically described in BEC phenomenon. In this paper, we prove the existence of the general exited states by use of the finite dimensional reduction method combined with the local Pohozaev identities. Precisely, we construct vector peak solutions as a1+a2−2βa1a2−β2→1ka⁎, where a1,a2>0,β∈(−a1a2,min⁡{a1,a2})∪(max⁡{a1,a2},+∞), k is a positive integer, a⁎=∫R2W2 and W is the unique positive solution of −Δw+w=w3 in R2. Our results generalize those obtained in [9], where Guo-Li-Wei-Zeng proved the existence of the ground states for fixed 0<a1,a2≤a⁎ and β>0 by variational methods.Compared with the single equation, the difficulties in the study of BEC systems come from the interspecies interaction between the components, which requires subtle estimates.

Non-degeneracy of bubbling solutions for fractional Schrödinger equation and its application

We consider the following fractional Schrödinger equation:(−Δ)su+V(y)u=u2s⁎−1, in RN, where 12≤s<1, 2s⁎=2NN−2s is the critical Sobolev exponent, and V(y) is a positive function. By using four types of local Pohozaev identities, we prove a non-degeneracy result of the positive bubbling solutions. We also give an application of the non-degeneracy result.

Non-degeneracy of bubble solutions for higher order prescribed curvature problem

Abstract In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator on S N {{\mathbb{S}}}^{N} : D m u = K ( ∣ y ∣ ) u m ∗ − 1 , u &gt; 0 in S N , u ∈ H m ( S N ) , {D}^{m}u=K\left(| y| ){u}^{{m}^{\ast }-1},\hspace{1.0em}u\gt 0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{S}}}^{N},\hspace{1.0em}u\in {H}^{m}\left({{\mathbb{S}}}^{N}), where K ( ∣ y ∣ ) K\left(| y| ) is a positive function, m ∗ = 2 N N − 2 m {m}^{\ast }=\frac{2N}{N-2m} is the Sobolev embedding critical exponent, N &gt; 2 m + 2 N\gt 2m+2 . D m {D}^{m} is the 2 m 2m order differential operator given by D m = ∏ l = 1 m − Δ g + 1 4 ( N − 2 l ) ( N + 2 l − 2 ) , {D}^{m}=\mathop{\prod }\limits_{l=1}^{m}\left(-{\Delta }_{g}+\frac{1}{4}\left(N-2l)\left(N+2l-2)\right), where Δ g {\Delta }_{g} is the Laplace-Beltrami operator on S N {{\mathbb{S}}}^{N} , S N {{\mathbb{S}}}^{N} is the unit sphere with Riemann metric g g . We first establish two kinds of local Pohozaev identities for polyharmonic operator, then we prove that the positive bubbling solution constructed in the study of Guo and Li is non-degenerate.

Cylindrical Solutions for a Critical Grushin-Type Equation via Local Pohozaev Identities

Cylindrical Solutions for a Critical Grushin-Type Equation via Local Pohozaev Identities

Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

We are concerned with the Lane-Emden problem{−Δu=upinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂R2 is a smooth bounded domain and p>1 is sufficiently large.Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.

Local uniqueness of multi-bump solutions for singularly perturbed Kirchhoff problems

We consider the following Kirchhoff type problem: −ɛ2a+ɛb∫R3|∇u|2Δu+V(x)u=|u|p−2u,x∈R3,where a,b>0, p∈(4,6), V∈C1(R,[0,∞)) and ɛ>0 is a parameter. Under some local assumptions on V, we prove the uniqueness of multi-bump solutions concentrating around zero points of V by a local Pohozaev identity.

Non-degeneracy of single-peak solutions to a Kirchhoff equation

ABSTRACT In this paper, we are concerned with the following singular perturbation Kirchhoff equation where constants and 1<p<5. By using local Pohozaev identity and blow-up analysis, we derive the non-degeneracy of the single-peak solutions to the above problem under some suitable condition on for small ε.

A singular perturbed problem with critical Sobolev exponent

This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u&gt; 0\quad\text{in}\ \R^N, \end{equation} where $\omega\in\R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon&gt; 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.

Uniqueness of single-peaked solutions to singularity perturbed semilinear Dirichlet problems

We are concerned with the uniqueness of single-peaked solutions to the following problem −ε2△u+u=Q(x)up−1,inΩ,u>0,inΩ,u=0,on∂Ω,where ε>0 is a small parameter, and N≥3, 2<p<2N∕(N−2). By local Pohozaev identity and blow-up analysis, we show the uniqueness of single-peaked solutions under certain assumptions on asymptotic behavior of Q(x) and its first derivatives near the critical point. Here the degeneracy of the critical point is also allowed, which gives a partial answer to the question proposed in Cao et al. (1998).