Ljusternik Schnirelmann Category Theory Research Articles

Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

This paper deals with the following fractional Schrödinger–Poisson system: [Formula: see text] with multiple competing potentials and a critical nonlocal term, where [Formula: see text], [Formula: see text] or [Formula: see text], and [Formula: see text] is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient [Formula: see text] of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of [Formula: see text].

Multiplicity of positive solution for Schrödinger-Poisson system with P-Laplacian

In this paper, we consider the positive solutions for a Schrödinger-Poisson system containing p-Laplacian and Poisson equation. With a local condition on the potential, by using penalization method, Nehari manifold and Ljusternik-Schnirelmann category theory, the multiplicity and concentration of solutions are obtained.

Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction

This paper focuses on the study of multiplicity and concentration phenomena of positive solutions for the singularly perturbed double phase problem with nonlocal Choquard reaction{−ϵpΔpu−ϵqΔqu+V(x)(|u|p−2u+|u|q−2u)=ϵμ−N(1|x|μ⁎G(u))g(u),inRN,u∈W1,p(RN)∩W1,q(RN),u>0,inRN, where 1<p<q<N, 0<μ<N, ϵ is a small positive parameter and V is the absorption potential. Combining variational and topological arguments from Nehari manifold analysis and Ljusternik-Schnirelmann category theory, we prove the existence of positive ground state solutions that concentrate around global minimum points of the potential V. In the second part of this paper, we establish the relationship between the number of positive solutions and the topology of the set where V attains its global minimum. The main results included in this paper complement several recent contributions to the study of concentration phenomena.

Fractional (p, q)-Schrödinger Equations with Critical and Supercritical Growth

In this paper, we complete the study started in Ambrosio and Rădulescu (J Math Pures Appl (9) 142:101–145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrödinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional (p, q)-Laplacian problems: (-Δ)psu+(-Δ)qsu+V(εx)(up-1+uq-1)=f(u)+uqs∗-1inRN,u∈Ws,p(RN)∩Ws,q(RN),u>0inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} (-\\Delta )^{s}_{p}u+(-\\Delta )^{s}_{q}u + V(\\varepsilon x) (u^{p-1} + u^{q-1})= f(u)+u^{q^{*}_{s}-1} \\, \ ext{ in } \\mathbb {R}^{N}, \\\\ u\\in W^{s, p}(\\mathbb {R}^{N})\\cap W^{s,q}(\\mathbb {R}^{N}), \\, u>0 \ ext{ in } \\mathbb {R}^{N}, \\end{array} \\right. \\end{aligned}$$\\end{document}where varepsilon >0 is a small parameter, sin (0, 1), 1<p<q<frac{N}{s}, q^{*}_{s}=frac{Nq}{N-sq} is the fractional critical Sobolev exponent, (-Delta )^{s}_{r}, with rin {p, q}, is the fractional r-Laplacian operator, V:mathbb {R}^{N}rightarrow mathbb {R} is a positive continuous potential such that inf _{partial Lambda }V>inf _{Lambda } V for some bounded open set Lambda subset mathbb {R}^{N}, and f:mathbb {R}rightarrow mathbb {R} is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik–Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when f(t)=t^{gamma -1}+mu t^{tau -1}, with 1< p<q<gamma<q^{*}_{s}<tau and mu >0 sufficiently small, by combining a truncation argument with a Moser-type iteration.

Semi-classical solutions for generalized quasilinear Schrödinger equations with nonlocal term and upper critical exponent

In this paper, we focus on the following generalized quasilinear Schrödinger equation with nonlocal term and critical growth: where , , , , , is a bounded even function, , for all , for some and for all . By using the variational methods and Ljusternik–Schnirelmann category theory, we establish the existence and multiplicity of semi-classical solutions.

Multiplicity and concentration results for a (p, q)-Laplacian problem in $${\mathbb {R}}^{N}$$

In this paper, we study the multiplicity and concentration of positive solutions for the following (p, q)-Laplacian problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb {R}}^{N}, \\ u\in W^{1, p}({\mathbb {R}}^{N})\cap W^{1, q}({\mathbb {R}}^{N}), \quad u>0 \text{ in } {\mathbb {R}}^{N}, \end{array} \right. \end{aligned}$$where \(\varepsilon >0\) is a small parameter, \(1< p<q<N\), \(\Delta _{r}u={{\,\mathrm{div}\,}}(|\nabla u|^{r-2}\nabla u)\), with \(r\in \{p, q\}\), is the r-Laplacian operator, \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a continuous function satisfying the global Rabinowitz condition, and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik–Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small \(\varepsilon \).

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 of concentrating positive solutions for nonlinear Kirchhoff type problems with critical growth

ABSTRACT In this paper, we study the following semilinear Kirchhoff type equation: where is a parameter, a, b are positive constants, V and f are continuous functions. Under certain assumptions on V and f, we investigate the relation between the number of solutions and the topology of the set where V attains its local minimum for small ε. We also describe the concentration phenomena of solutions as . The proof is based on the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.

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.

On the multiplicity and concentration of positive solutions for a [formula omitted]-fractional Choquard equation in [formula omitted

In this paper we deal with the following fractional Choquard equation εsp(−Δ)psu+V(x)|u|p−2u=εμ−N1|x|μ∗F(u)f(u)inRN,u∈Ws,p(RN),u>0inRN,where ε>0 is a small parameter, s∈(0,1), p∈(1,∞), N>sp, (−Δ)ps is the fractional p-Laplacian, V is a positive continuous potential, 0<μ<sp, and f is a continuous superlinear function with subcritical growth. Using minimax arguments and the Ljusternik–Schnirelmann category theory, we obtain the existence, multiplicity and concentration of positive solutions for ε>0 small enough.

On a fractional magnetic Schrödinger equation in [formula omitted] with exponential critical growth

We deal with a class of fractional magnetic Schrödinger equations in the whole line with exponential critical growth. Under a local condition on the potential, we use penalization methods and Ljusternik–Schnirelmann category theory to investigate the existence, multiplicity and concentration of nontrivial weak solutions.

On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity

In the present paper we deal with a nonlinear fractional Schrödinger equation in of the formwhere is a parameter, , , V(x) is a continuous potential, and is the Riesz potential. By applying variational methods and the Ljusternik–Schnirelmann category theory, under some suitable conditions on V(x) we prove the existence and multiplicity of solutions for the above equation when the parameter is sufficiently small. Moreover, the concentration behavior of solutions for the fractional Schrödinger equation is also explored.

Multiple positive solutions for a class of Kirchhoff type problems involving critical Sobolev exponents

In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems involving critical growth terms. With the help of Nehari manifold and Ljusternik–Schnirelmann category theory, we investigate how the coefficient g(x) of the critical nonlinearity affects the number of positive solutions of that problem. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g.

Existence and multiplicity results for perturbed Kirchhoff-type Schrödinger systems in [formula omitted

In this paper we study a class of Kirchhoff-type Schrödinger systems with a positive parameter ε. By applying variational methods and the Ljusternik–Schnirelmann category theory, under some suitable conditions we obtain the existence and multiplicity results for the systems when the parameter ε is sufficiently small.

Periodic solutions for a class of nonlinear discrete Hamiltonian systems via critical point theory

In this paper, we study the multiple existence of periodic solutions of the discrete n-dimensional Hamiltonian system with V periodic both in t and the components of q. By showing that the action functional of this Hamiltonian system is invariant under the action of the non-compact group , we construct a new functional for which the Palais-Smale condition holds and prove multiple existence of periodic solutions of this system via Ljusternik–Schnirelmann category theory and Morse theory.

Existence and multiplicity results for massive particles trajectories in a universe with boundary

In an open time-convex region Λ of a strongly causal Lorentzian manifold (M,g), we consider an event p and a timelike, injective curve γ. We look for geodesics connecting p and γ in Λ and satisfying the conservation law g(z)[ż,ż]=−E for a fixed E&amp;gt;0. It is already known that such geodesics are the stationary points of the arrival time functional τ. Our main result is to prove the existence of a decreasing flow for τ, by means of a shortening procedure. This makes possible to apply to τ global variational methods obtaining existence and multiplicity results (using the Ljusternik–Schnirelmann category theory) and also to develop a Morse theory.

Periodic Orbits and Subharmonics of Dynamical Systems on Non-Compact Riemannian Manifolds

where Dt(x* (t)) is the covariant derivative of x* along the direction of x* and {R the Riemannian gradient, has been studied when M is a noncontractible manifold (see [2, 8, 9, 14]), assuming, if M is non-compact, the existence of a function on M convex at infinity . When V is bounded the difficulties arise from the lack of compactness of M; indeed, in this case the action functional does not satisfy the Palais Smale compactness condition. On the other side, if V is unbounded the action functional is unbounded both from below and from above. Therefore neither min max methods and linking arguments can be used since the loop space is not linear nor the Ljusternik Schnirelmann category theory can be applied although M has a non-trivial topology. article no. 0136

Extensions of Ljusternik-Schnirelmann category theory to relative and equivariant theories with an application to an equivariant critical point theorem

We extend Ljusternik-Schnirelmann category theory to a relative G-category theory. The theory is related to the equivariant cohomological index of Fadell and Husseini. After giving computations for a number of examples, we present an application of the theory to the problem of finding critical points of invariant functionals.