Ljusternik-Schnirelmann Theory Research Articles

Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2

In this article, we study the generalized Kadomtsev-Petviashvili equation witha potential $$ (-u_{xx}+D_{x}^{-2}u_{yy}+V(\varepsilon x,\varepsilon y)u-f(u))_{x}=0\quad \text{in }\mathbb{R}^2, $$ where \(D_{x}^{-2}h(x,y)=\int_{-\infty }^{x}\int_{-\infty }^{t}h(s,y)\,ds\,dt \), \(f\) is a nonlinearity, \(\varepsilon\) is a small positive parameter, and the potential \(V\) satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by applying penalization techniques. The relationship between the number of positive solutions and the topology of the set where \(V\) attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser's iteration method, we verify that the solutions of the modified problem are indeed solutions of the original roblem for \(\varepsilon>0\) small enough.

On a class of fractional Kirchhoff–Schrödinger–Poisson systems involving magnetic fields

This article establishes the multiplicity and concentration of solutions for a fractional Kirchhoff–Schrödinger–Poisson systems with magnetic field (aɛ2s+bɛ4s−3[u]A/ɛ2)(−Δ)A/ɛsu+V(x)u+ϕ(x)u=f(|u|2)uinR3,ɛ2t(−Δ)tϕ=|u|2inR3,where ɛ>0 is a small parameter, both of V:R3→R and A:R3→R3 are continuous potential and magnetic potential, and f∈C1(R,R) is a subcritical nonlinearity. More precisely, we explore how multiplicity results for solutions of the aforementioned system are linked to the topology of the set where V attains the minimum and how concentration results of solutions changes as parameter ɛ changes. The key tools are variational approaches, penalization technique as well as Ljusternik–Schnirelmann theory. As far as we know, our results extends in several directions previous results for fractional Schrödinger–Poisson systems.

Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations

Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{\varepsilon }^{2}\Delta u+V\left(x)u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }\ast | u{| }^{2})u=\lambda | u{| }^{p-1}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where ε , λ > 0 \varepsilon ,\lambda \gt 0 are parameters, N ⩾ 2 N\geqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 \left(\alpha +6)\hspace{0.1em}\text{/}\hspace{0.1em}\left(\alpha +2)\lt p\lt {2}^{\ast }-1 , I α {I}_{\alpha } is the Riesz potential with 0 < α < N 0\lt \alpha \lt N , and V ∈ C ( R N , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{N},{\mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.

Multiplicity and concentration of solutions to fractional anisotropic Schrödinger equations with exponential growth

In this paper, we consider the Schrödinger equation involving the fractional (p,p_1,dots ,p_m)-Laplacian as follows (-Δ)psu+∑i=1m(-Δ)pisu+V(εx)(|u|(N-2s)/2su+∑i=1m|u|pi-2u)=f(u)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (-\\Delta )_{p}^{s}u+\\sum _{i=1}^{m}(-\\Delta )_{p_i}^{s}u+V(\\varepsilon x)(|u|^{(N-2s)/2s}u+\\sum _{i=1}^{m}|u|^{p_i-2}u)=f(u)\\;\ ext{ in }\\; {\\mathbb {R}}^{N}, \\end{aligned}$$\\end{document}where varepsilon is a positive parameter, N=ps, sin (0,1), 2le p<p_1< dots< p_m<+infty , mge 1. The nonlinear function f has the exponential growth and potential function V is continuous function satisfying some suitable conditions. Using the penalization method and Ljusternik–Schnirelmann theory, we study the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the above problem is studied.

The infimum eigenvalue for degenerate p(x)-biharmonic operator with the Hardy potentiel

The aim of this article is to study the existence of at least one unbounded nondecreasing sequence of nonnegative eigenvalues (λk)k≥1 for a class of elliptic Navier boundary value problems involving the degenerate p(·)-biharmonic operator with q(x)-Hardy inequality by using the variational technique based on the Ljusternik-Schnirelmann theory on C1-manifolds and the theory of the variable exponent Lebesgue spaces. Also, we obtain the positivity of the infimum eigenvalue for the problem.

Multiplicity and concentration results for fractional Kirchhoff equation with magnetic field

ABSTRACT In the paper, we considered the fractional Kirchhoff equation with magnetic field ( a ε 2 s + b ε 4 s − 3 [ u ] A / ε 2 ) ( − Δ ) A / ε s u + V ( x ) u = f ( | u | 2 ) u in R 3 , where ε is a small parameter, a, b are positive constants, s ∈ ( 3 4 , 1 ) , A : R 3 → R 3 is a Hölder continuous magnetic potential with exponent α ∈ ( 0 , 1 ] , [ u ] A / ε 2 = ∬ R 6 | u ( x ) − ε ı ( x − y ) ⋅ A ε ( x + y 2 ) u ( y ) | 2 | x − y | 3 + 2 s d x d y . With proper assumptions on V and f, we develop a new approach to prove the Palais–Smale condition for the corresponding energy functional and establish the multiplicity and concentration of solutions by using perturbation techniques and Ljusternik–Schnirelmann theory.

Ground state solutions of Schrödinger system with fractionalp-Laplacian

AbstractThis article deals with a class of nonlinear fractionalp-Laplacian Schrö$\ddot{o}$dinger coupled system with critical and subcritical nonlinear terms. Firstly, the existence of a nonnegative ground state solution of the system is proved by the Nehari manifold method and the Ekeland’s variational principle. In addition, through the Ljusternik–Schnirelmann theory, we link the number of solutions to the topology of the set in which the potentials in the system reach their minimum values.

Multiplicity and concentration of solutions to a fractional p-Laplace problem with exponential growth

&#x0D; In this paper, we study the Schrödinger equation involving \(\frac{N}{s}\)-fractional Laplace as follows&#x0D; \(\varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)|u|^{\frac{N}{s}-2}u=f(u)\) in \(\mathbb R^{N}\),&#x0D; where \(\varepsilon\) is a positive parameter, \(N=ps\), \(s\in (0,1)\). The nonlinear function \(f\) has the exponential growth and potential function \(V\) is a continuous function satisfying some suitable conditions. Our problem lacks of compactness. By using the Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter.&#x0D;

Multiple solutions for a fractional p&amp;q-Laplacian system involving Hardy-Sobolev exponent

In this paper, we prove the existence of infinitely many solutions for a fractional p&amp;q-Laplacian system involving Hardy-Sobolev exponents and obtain new conclusion under different conditions. The methods used here are based on variational methods and LjusternikSchnirelmann theory.

Normalized solutions to a Schrödinger–Bopp–Podolsky system under Neumann boundary conditions

In this paper, we study a Schrödinger–Bopp–Podolsky (SBP) system of partial differential equations in a bounded and smooth domain of [Formula: see text] with a nonconstant coupling factor. Under a compatibility condition on the boundary data we deduce existence of solutions by means of the Ljusternik–Schnirelmann theory.

Multiplicity and concentration of solutions to a fractional (p,p1)-Laplace problem with exponential growth

In this paper, we study the Schrödinger equation involving the fractional (p,p1)-Laplacian and Trudinger-Moser nonlinearity as follows(−Δ)N/ssu+(−Δ)p1su+V(εx)(|u|Ns−2u+|u|p1−2u)=f(u)inRN, where ε is a positive parameter, N=ps,s∈(0,1),2≤Ns=p<p1. The nonlinear function f has the exponential growth and potential function V is continuous function satisfying some suitable conditions. By using the Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the fractional (p,p1)-Laplacian problem involving exponential growth is studied.

Fermat Metrics

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

Existence and concentration results for fractional Schrodinger-Poisson system via penalization method

This article concerns the positive solutions for the fractional Schrodinger-Poisson system $$\displaylines{ \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)\quad \text{in }\mathbb{R}^3,\cr \varepsilon^{2t}(-\Delta)^t\phi=u^2\quad \text{in }\mathbb{R}^3, }$$ where \(\varepsilon&gt;0\) is a small parameter, \((-\Delta)^\alpha\) denotes the fractional Laplacian of orders \(\alpha=s,t\in(3/4,1)\), \(V\in C(\mathbb{R}^3,\mathbb{R})\) is the potential function and \(f:\mathbb{R}\to\mathbb{R}\) is continuous and subcritical. Under a local condition imposed on the potential function, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values. Moreover, we considered some properties of these positive solutions, such as concentration behavior and decay estimate. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/14/abstr.html

Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

Abstract This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C 1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ 1(λ) and also show that, the smallest curve μ 1(λ) is positive for all 0 ≤ λ &lt; C H , with C H is the optimal constant of Hardy type inequality.

Multiplicity and concentration of nontrivial solutions for fractional Schrödinger–Poisson system involving critical growth

In this paper, we study the concentration and multiplicity of solutions to the following fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u+ϕu=f(u)+u2s∗−1in R3, ε2t(−Δ)tϕ=u2,u>0in R3, where s,t∈(0,1), s>34, ε>0 is a small parameter, f∈C1(R+,R), and V:R3→R is a continuous and bounded function satisfying some local assumptions. We establish a family of positive solutions uε∈Hε which concentrates around the set of local minima of V as ε→0. Additionally, through using the Ljusternik–Schnirelmann theory, we also obtain the multiplicity of solutions by employing the topology of the set where the potential V attains its minimum.

Periodic Solutions for N-Body-Type Problems

The authors consider non-autonomous N-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity. Using Ljusternik-Schnirelmann theory, the authors prove the existence of unbounded sequences of critical values for the Lagrangian action corresponding to non-collision periodic solutions.

Multiplicity of positive solutions for a fractional p&q-Laplacian problem in [formula omitted

In this paper we deal with the following fractional p&q-Laplacian problem:{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, where s∈(0,1), ε>0 is a small parameter, 2≤p<q<Ns, (−Δ)ts, with t∈{p,q}, is the fractional (s,t)-Laplacian operator, V:RN→R is a continuous function satisfying the global Rabinowitz condition, and f:R→R is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we prove that the above problem admits multiple solutions for ε>0 small enough.

Multiplicity and concentration of solutions to the nonlinear magnetic Schr\xf6dinger equation

In this paper, we study the following nonlinear magnetic Schrödinger equation (εi∇-A(x))2u+V(x)u=f(|u|2)uinRN(N≥2),u∈H1(RN,C),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned}&\\Big (\\frac{\\varepsilon }{i}\\nabla -A(x)\\Big )^{2}u+V(x)u=f(|u|^{2})u \\quad \\hbox {in }{\\mathbb {R}}^N\\ (N\\ge 2),\\\\&u\\in H^{1}({\\mathbb {R}}^{N}, {\\mathbb {C}}), \\end{aligned} \\right. \\end{aligned}$$\\end{document}where epsilon is a positive parameter, and V:{mathbb {R}}^{N}rightarrow {mathbb {R}}, A: {mathbb {R}}^{N}rightarrow {mathbb {R}}^{N} are continuous potentials. Under a local assumption on the potential V, by combining variational methods, penalization techniques, and the Ljusternik–Schnirelmann theory, we prove multiplicity and concentration properties of solutions for varepsilon >0 small. In our problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Multiplicity and Concentration Results for a Magnetic Schrödinger Equation With Exponential Critical Growth in ℝ2

AbstractIn this paper we study the following nonlinear Schrödinger equation with magnetic field $$\begin{align*} \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \end{align*}$$where $\varepsilon&amp;gt;0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials, and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternik–Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon $ small.

Existence and multiplicity of positive solutions for Kirchhoff-Schrödinger-Poisson system with critical growth

This paper is concerned with the following Kirchhoff-Schrodinger-Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+ b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u + V(x)u+\phi u=\lambda g(x)u^{q-1}+h(x)u^{5}, &{} \ \ \text{ in } \mathbb {R}^{3},\\ -\Delta \phi = u^{2},~u>0,&{} \ \ \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$where $$a>0$$, $$b\ge 0$$, $$q\in [4,6)$$ and $$\lambda >0$$ is a parameter. Under some suitable conditions on V(x), g(x) and h(x), by using the Nehari manifold technique and the Ljusternik-Schnirelmann category theory, we relate the number of positive solutions with the topology of the global maximum set of h when $$\lambda $$ is small enough. Furthermore, with the aid of the Mountain Pass Theorem, we also obtain an existence result for $$\lambda $$ sufficiently large. Recent results from the literature are generally improved and extended.