Positive Energy Solutions Research Articles

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities

Abstract We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by S u ≔ − div ϕ u 2 + ∣ ∇ u ∣ 2 2 ∇ u + ϕ u 2 + ∣ ∇ u ∣ 2 2 u , Su:= -\hspace{0.1em}\text{div}\hspace{0.1em}\left\{\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)\nabla u\right\}+\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)u, where ϕ : [ 0 , + ∞ ) → R \phi :\left[0,+\infty )\to {\mathbb{R}} is a positive continuous function. This operator is introduced by Stuart [Two positive solutions of a quasilinear elliptic Dirichlet problem, Milan J. Math. 79 (2011), 327–341] and depends on not only ∇ u \nabla u but also u u . This particular quasilinear term generally appears in the study of nonlinear optics model, which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution, which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S S : (i) the almost everywhere convergence of the gradient for the minimizing sequence { u n } \left\{{u}_{n}\right\} ; (ii) seeking the reasonable restrictions about S S .

Multiple high energy solutions of a nonlinear Hardy–Sobolev critical elliptic equation arising in astrophysics

In this article, we study the existence and multiplicity of high energy solutions to the problem proposed as a model for the dynamics of galaxies: −Δu+V(x)u=|u|2∗−2u|y|,x=(y,z)∈Rm×Rn−m,where n>4, 2≤m<n, 2∗≔2(n−1)n−2 and potential function V(x):Rn→R. Benefiting from a global compactness result, we show that there exist at least two positive high energy solutions. Our proofs are based on barycenter function, quantitative deformation lemma and Brouwer degree theory.

Observing Majorana fermion dynamic properties on a NISQ computer

Majorana particles are of particular interest due to their profound property of being their own anti-particles and potential applications for quantum computers. However, studying the dynamics of Majorana particles experimentally has proven to be challenging, with many unanswered questions about their behaviors. In this paper, we demonstrate that an improved discrete-time quantum walks (DTQW) on a Noisy Intermediate-Scale Quantum (NISQ) computer can be used to construct a well-defined Majorana fermion. Through this approach, we can observe the behaviors of Majorana particle propagation both in (1 + 1)- and (3 + 1)-dimensions. A counter-intuitive dynamic behavior of the non-vanishing expectation value of velocity and position with zero expectation value of momentum from the superposition of positive and negative energy solution is observed for the first time in NISQ. Our findings provide a promising avenue for experimentally studying the dynamical properties of Majorana particles.

High Energy Solutions for p-Kirchhoff Elliptic Problems with Hardy–Littlewood–Sobolev Nonlinearity

This article deals with the study of the following Kirchhoff–Choquard problem: M∫RN|∇u|p(-Δp)u+V(x)|u|p-2u=∫RNF(u)(y)|x-y|μdyf(u),inRN,u>0,inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\begin{array}{cc} \\displaystyle M\\left( \\, \\int \\limits _{{\\mathbb {R}}^N}|\ abla u|^p\\right) (-\\Delta _p) u + V(x)|u|^{p-2}u = \\left( \\, \\int \\limits _{{\\mathbb {R}}^N}\\frac{F(u)(y)}{|x-y|^{\\mu }}\\,dy \\right) f(u), \\;\\;\ ext {in} \\; {\\mathbb {R}}^N,\\\\ u > 0, \\;\\; \ ext {in} \\; {\\mathbb {R}}^N, \\end{array} \\end{aligned}$$\\end{document}where M models Kirchhoff-type nonlinear term of the form M(t)=a+btθ-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(t) = a + bt^{\ heta -1}$$\\end{document}, where a,b>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a, b > 0$$\\end{document} are given constants; 1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<N$$\\end{document}, Δp=div(|∇u|p-2∇u)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _p = \ ext {div}(|\ abla u|^{p-2}\ abla u)$$\\end{document} is the p-Laplacian operator; potential V∈C2(RN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V \\in C^2({\\mathbb {R}}^N)$$\\end{document}; f is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for θ∈1,2N-μN-p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in \\left[ 1, \\frac{2N-\\mu }{N-p}\\right) $$\\end{document} via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.

Existence and classification of positive solutions for coupled purely critical Kirchhoff system

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

A logarithmic Choquard equation with critical exponential growth in ℝ2

ABSTRACT In this paper, we are paid attention to the following logarithmic Choquard equation − Δ u + u = [ I 2 ∗ F ( u ) ] f ( u ) , x ∈ R 2 , where I 2 = 1 2 π ln ⁡ 1 | x | is the Newtonian kernel in dimension 2, F ( u ) = ∫ 0 u f ( t ) d t and f ∈ C 1 ( R , R ) is critical exponential growth with respect to Trudinger–Moser. Based on a variational framework proposed in Liu et al. (J Differ Equ 2022;328), we obtain a positive finite energy solution of the above logarithmic Choquard equation via the variational methods. Furthermore, we use another approach to investigate the planar logarithmic Choquard equations involving critical exponential growth.

Vacuum Effects Induced by a Plate in de Sitter Spacetime in the Presence of a Cosmic String

In this paper, we investigate the vacuum expectation values of the field squared and the energy–momentum tensor associated to a charged massive scalar quantum field in a (1+D)-dimensional de Sitter spacetime induced by a plate (flat boundary) and a carrying-magnetic-flux cosmic string. In our analysis, we admit that the flat boundary is perpendicular to the string, and the scalar field obeys the Robin boundary condition on the plate. In order to develop this analysis, we obtain the complete set of normalized positive-energy solutions of the Klein–Gordon equation compatible with the model setup. Having obtained these bosonic modes, we construct the corresponding Wightman function. The latter is given by the sum of two terms: one associated with the boundary-free spacetime, and the other induced by the flat boundary. Although we have imposed the Robin boundary condition on the field, we apply our formalism considering specifically the Dirichlet and Neumann boundary conditions. The corresponding parts have opposite signs. Because the analysis of bosonic vacuum polarization in boundary-free de Sitter space and in the presence of a cosmic string, in some sense, has been developed in the literature, here we are mainly interested in the calculations of the effects induced by the boundary. In this way, closed expressions for the corresponding expectation values are provided, as well as their asymptotic behavior in different limiting regions. We show that the conical topology due to the cosmic string enhances the boundary-induced vacuum polarization effects for both field squared and the energy–momentum tensor, compared to the case of a boundary in pure de Sitter spacetime. Moreover, the presence of a cosmic string and boundary induces non-zero stress along the direction normal to the boundary. The corresponding vacuum force acting on the boundary is also investigated.

A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: Existence and multiplicity of high energy positive solutions

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent{−Δu+(V1(x)+λ1)u=μ1(|x|−4⁎u2)u+β(|x|−4⁎v2)u,x∈RN,−Δv+(V2(x)+λ2)v=μ2(|x|−4⁎v2)v+β(|x|−4⁎u2)v,x∈RN, where N≥5, λ1, λ2≥0 and λ1+λ2≠0, V1(x),V2(x)∈LN2(RN) are nonnegative functions and μ1, μ2, β are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of high energy positive solutions under the hypothesis β>max⁡{μ1,μ2}.

Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

Infinitely many positive energy solutions for an elliptic equation involving critical Sobolev growth, Hardy potential and concave–convex nonlinearity

Infinitely many positive energy solutions for an elliptic equation involving critical Sobolev growth, Hardy potential and concave–convex nonlinearity

Bound state positive solutions for a Hartree system with nonlinear couplings

In this paper, we are interested in the following Hartree system with nonlinear couplings: { − ε 2 Δ u + V 1 ( x ) u = 1 ε N − μ [ ν 1 ( ∫ R N | u | p | x − y | μ d y ) | u | p − 2 u + β ( ∫ R N | v | q | x − y | μ d y ) | u | q − 2 u ] , − ε 2 Δv + V 2 ( x ) v = 1 ε N − μ [ ν 2 ( ∫ R N | v | p | x − y | μ d y ) | v | p − 2 v + β ( ∫ R N | u | q | x − y | μ d y ) | v | q − 2 v ] , u , v ∈ H 1 ( R N ) , u , v > 0 in R N , where N ≥ 3 , 0 < μ < N , 2 N − μ N ≤ p ≤ 2 N − μ N − 2 , 2 N − μ N ≤ q ≤ min { p , 2 } , ν 1 , ν 2 > 0 , ε is a small parameter and β < 0 is a coupling constant, and the potentials V 1 and V 2 have k 1 and k 2 isolated global minimum points, respectively. Using the Nehari manifold technique, the energy estimate method and the Lusternik–Schnirelmann theory, we find an interesting phenomenon that the problem possesses k 1 k 2 positive solutions when V 1 and V 2 do not have any common isolated global minimum points, and k 1 k 2 + m positive solutions when V 1 and V 2 have m common isolated global minimum points. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.

Many-body theories for negative kinetic energy systems

In the author’s previous works, it is derived from the Dirac equation that particles can have negative kinetic energy (NKE) solutions, and they should be treated on an equal footing as the positive kinetic energy (PKE) solutions. More than one NKE particles can make up a stable system by means of interactions between them and such a system has necessarily negative temperature. Thus, many-body theories for NKE systems are desirable. In this work, the many-body theories for NKE systems are presented. They are Thomas‐Fermi method, Hohenberg‐Kohn theorem, Khon‐Sham self-consistent equations, and Hartree‐Fock self-consistent equations. They are established imitating the theories for PKE systems. In each theory, the formalism of both zero temperature and finite negative temperature is given. In order to verify that tunneling electrons are of NKE and real momentum, an experiment scenario is suggested that lets PKE electrons collide with tunneling electrons.

Least energy positive solutions for d-coupled Schrödinger systems with critical exponent in dimension three

In the present paper, we consider the coupled Schrödinger systems with critical exponent:{−Δui+λiui=∑j=1dβij|uj|3|ui|ui in Ω,ui∈H01(Ω),i=1,2,...,d. Here, Ω⊂R3 is a smooth bounded domain, d≥2, βii>0 for every i, and βij=βji for i≠j. We study a Brézis-Nirenberg type problem: −λ1(Ω)<λ1,⋯,λd<−λ⁎(Ω), where λ1(Ω) is the first eigenvalue of −Δ with Dirichlet boundary conditions and λ⁎(Ω)∈(0,λ1(Ω)). We acquire the existence of least energy positive solutions to this system for weakly cooperative case (βij>0 small) and for purely competitive case (βij≤0) by variational arguments. The proof is performed by mathematical induction on the number of equations, and requires more refined energy estimates for this system. Besides, we present a new nonexistence result, revealing some different phenomena comparing with the higher-dimensional case N≥5. It seems that this is the first paper to give a rather complete picture for the existence of least energy positive solutions to critical Schrödinger system in dimension three.

Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity

Let Ω \Omega be a smooth bounded domain in R n \mathbb {R}^n ( n ≥ 3 n\geq 3 ) such that 0 ∈ ∂ Ω 0\in \partial \Omega . We consider issues of non-existence, existence, and multiplicity of variational solutions in H 1 , 0 2 ( Ω ) H_{1,0}^2(\Omega ) for the borderline Dirichlet problem, { − Δ u − γ u | x | 2 − h ( x ) u a m p ; = a m p ; | u | 2 ⋆ ( s ) − 2 u | x | s a m p ; in Ω , u a m p ; = a m p ; 0 a m p ; on ∂ Ω ∖ { 0 } , \begin{equation*} \left \{ \begin {array}{llll} -\Delta u-\gamma \frac {u}{|x|^2}- h(x) u &amp;=&amp; \frac {|u|^{2^\star (s)-2}u}{|x|^s} \ \ &amp;\text {in } \Omega ,\\ \hfill u&amp;=&amp;0 &amp;\text {on }\partial \Omega \setminus \{ 0 \} , \end{array} \right . \end{equation*} where 0 &gt; s &gt; 2 0&gt;s&gt;2 , 2 ⋆ ( s ) ≔ 2 ( n − s ) n − 2 {2^\star (s)}≔\frac {2(n-s)}{n-2} , γ ∈ R \gamma \in \mathbb {R} and h ∈ C 0 ( Ω ¯ ) h\in C^0(\overline {\Omega }) . We use sharp blow-up analysis on—possibly high energy—solutions of corresponding subcritical problems to establish, for example, that if γ &gt; n 2 4 − 1 \gamma &gt;\frac {n^2}{4}-1 and the principal curvatures of ∂ Ω \partial \Omega at 0 0 are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions in H 1 , 0 2 ( Ω ) H_{1,0}^2(\Omega ) . This complements results of the first and third authors, who showed in their 2016 article, Hardy-Singular Boundary Mass and Sobolev-Critical Variational Problems, that if γ ≤ n 2 4 − 1 4 \gamma \leq \frac {n^2}{4}-\frac {1}{4} and the mean curvature of ∂ Ω \partial \Omega at 0 0 is negative, then ( E ) (E) has a positive least energy solution. On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at 0 0 is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under C 1 C^1 -perturbations of the potential h h . In particular, and in sharp contrast with the non-singular case (i.e., when γ = s = 0 \gamma =s=0 ), we prove non-existence of such solutions for ( E ) (E) in any dimension, whenever Ω \Omega is star-shaped and h h is close to 0 0 , which include situations not covered by the classical Pohozaev obstruction.

High energy positive solutions for a coupled Kirchhoff system

High energy positive solutions for a coupled Kirchhoff system

Ground states of nonlinear Schrödinger systems with mixed couplings: the critical case

In this paper, we consider the following k-coupled nonlinear Schrödinger systems in the critical case: Here, is a smooth bounded domain, and for every , where is the first eigenvalue of with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (i.e. when ). We call the couplings are attractive if , while repulsive stands for . Under the assumption that all the couplings are purely attractive and large enough, we first show that this critical system has a fully nontrivial ground state solution, that is, a solution has all components nontrivial, under conditions providing additional , while ground state solution may be semitrivial ( has null components) without the above additional conditions. When the systems admit mixed couplings, i.e. there exist and such that and , we establish the existence of least energy positive solutions. The purpose of this paper is to solve some remaining open questions on Tavares and You (Calc. Var. Partial Differential Equations 59:26, 2020).

Existence and Multiplicity of Solutions for a Bi-Non-Local Problem

The aim of this paper is to investigate the existence and multiplicity of solutions for a bi-non-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help of the fountain theorem, we obtain the existence of infinitely many positive energy solutions, assuming a symmetric condition for g. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.

Combined effects of concave and convex nonlinearities for the generalized Chern–Simons–Schrödinger systems with steep potential well and 1 &amp;lt; p &amp;lt; 2 &amp;lt; q &amp;lt; 6

In this paper, we study the Chern–Simons–Schrödinger system with a steep potential well and 1 &amp;lt; p &amp;lt; 2 &amp;lt; q &amp;lt; 6. First, by using the truncation technique, we prove that this system possesses a positive energy solution. Second, the concentration behavior of the positive energy solutions as λ → +∞ and κ → 0 are also considered. Finally, we obtain a negative energy solution via the Ekeland variational principle.

Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: The higher dimensional case

Let Ω⊂RN be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with d≥2 equations−Δui+λiui=|ui|p−2ui∑j=1dβij|uj|p in Ω,ui=0 on ∂Ω,i=1,...,d, in the case of a critical exponent 2p=2⁎=2NN−2 in high dimensions N≥5. We treat the focusing case (βii>0 for every i) in the variational setting βij=βji for every i≠j, dealing with a Brézis-Nirenberg type problem: −λ1(Ω)<λi<0, where λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)). We provide several sufficient conditions on the coefficients βij that ensure the existence of least energy positive solutions; these include the situations of pure cooperation (βij>0 for every i≠j), pure competition (βij≤0 for every i≠j) and coexistence of both cooperation and competition coefficients.In order to provide these results, we firstly establish the existence of nonnegative solutions with 1≤m≤d nontrivial components by comparing energy levels of the system with those of appropriate sub-systems. Moreover, based on new energy estimates, we obtain the precise asymptotic behaviors of the nonnegative solutions in some situations which include an analysis of a phase separation phenomena. Some proofs depend heavily on the fact that 1<p<2, revealing some different phenomena comparing to the special case N=4. Our results provide a rather complete picture in the particular situation where the components are divided in two groups. Besides, based on the results about the phase separation, we prove the existence of least energy sign-changing solution to the Brézis-Nirenberg problem−Δu+λu=μ|u|2⁎−2u,u∈H01(Ω), for μ>0, −λ1(Ω)<λ<0 for all N≥4, a result which is new in dimensions N=4,5.

Ground state and least positive energy solutions of elliptic problems involving mixed fractional $p$-Laplacians

We establish the existence of ground state and least positive energy solutions to some problems involving mixed fractional $p$-Laplacians using some abstract critical point theorems based on the $\mathbb Z_2$-cohomological index.