Nehari Manifold Method Research Articles

Multiplicity of solutions for p-Laplacian Kirchhoff problems with singular nonlinearity

In this paper, we prove the existence of solutions for [Formula: see text]-Laplacian Kirchhoff problems with singular nonlinearity. More precisely, we use the Nehari manifold method and the analysis of the fibering maps in order to prove the existence of nonnegative solutions for such problem.

Solvability of a Class of Fractional Advection–Dispersion Coupled Systems

The purpose of this study is to provide some criteria for the existence and multiplicity of solutions for a class of fractional advection–dispersion coupled systems with nonlinear Sturm–Liouville conditions and instantaneous and non-instantaneous impulses. Specifically, the existence is derived through the Nehari manifold method, and the proof of multiplicity is based on Bonanno and Bisci’s critical point theorem, which does not require proof that the functional satisfies the Palais–Smale condition. Finally, to illustrate the obtained results, an example is provided.

Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

Ground state solutions for weighted biharmonic Kirchhoff problem via the Nehari manifold approach

In this article, we delve into the investigation of the following non-local problem within the unit ball B 1 of R 4 : g ( ∫ B 1 w β ( x ) | Δu | 2 ) Δ w β 2 u = | u | q − 2 u + f ( x , u ) in B 1 , u = ∂ n u = 0 on ∂ B 1 , where Δ w β 2 . = Δ ( w β ( x ) Δ . ) is the weighted Biharmonic operator, where w β ( x ) represents a logarithmically singular weight, and the non-linearity involves a combination of a reaction source f ( x , u ) , critical in the context of Adams' type exponential inequality and a polynomial function. The Kirchhoff function g is characterized by being positive and continuous. Employing the Nehari manifold method, the quantitative deformation lemma and degree theory results, we establish the existence of a ground state solution.

Traveling waves of a generalized sixth‐order Boussinesq equation

We investigate the existence and stability of traveling waves for the sixth‐order Boussinesq equation, considering a broad class of nonlinearities adhering to power‐like scaling relations. Employing the Nehari manifold method, we establish the existence of traveling waves and derive variational criteria for assessing their stability or instability. Subsequently, we develop a numerical approach based on these variational principles to delineate regions of stability and instability. Finally, for homogeneous nonlinearities, we establish a sufficient condition for strong instability.

On periodic and compactly supported least energy solutions to semilinear elliptic equations with non-Lipschitz nonlinearity

We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter λ, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p, q. Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 .

Periodic solutions for second-order even and noneven Hamiltonian systems

AbstractIn this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ x ¨ + V ′ ( x ) = 0 , x ∈ R N . We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

Multiplicity of solutions for a singular problem involving the fractional p$p$‐Laplacian in the whole space

AbstractIn this paper, we prove the multiplicity of positive solutions for the following singular problem involving the fractional ‐Laplacian: Here, , , and . Under appropriate assumptions on , , and , two positive entire solutions are obtained by using the Nehari manifold method.

Nehari manifold method for singular double phase problem with optimal control on parameter

In this paper, we consider the following singular double phase problem −div(|∇u|p−2∇u + a(x)|∇u|q−2∇u) = λf(x)u−γ + g(x)ur−1, u > 0 in Ω and u = 0 on ∂Ω, where Ω⊂RN is an open bounded domain with smooth boundary, dimension N ≥ 2, exponents p < q < r < p* = Np/(N − p) with 1 < p < N, while 0 < γ < 1 and λ > 0 is real parameter. The weight functions f, g are bounded continuous functions which may change sign and the modulating function a is non-negative, continuous and has compact support in Ω. Using fibering map and Nehari manifold method, we show the existence of at least two positive solutions for (0, λ* + ϵ) for some ϵ > 0, where λ* is an extremal parameter, characterized via nonlinear Rayleigh quotient. An estimate on the extremal value λ* is also obtained.

Singular two-phase problem on a complete manifold: analysis and insights

We focus on two-phase problems with singular and superlinear parametric terms on the right-hand side. Using fibering maps and the Nehari manifold method, we prove that there are at least two non-trivial positive solutions in a geometric setting that is locally similar to Euclidean spaces but has different global properties for all except the smallest values of parameter μ>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu > 0.$$\\end{document} Singularities may appear at discrete locations in the manifold, which is a challenge for the work due to the unpredictable behavior of the solution. The findings presented here generalize some known results.

On an elliptic Kirchhoff–Boussinesq type problems with exponential growth

In this paper, we prove an existence result of nontrivial solutions for the problem where is a smooth bounded domain, and is a superlinear continuous function with exponential subcritical or critical growth. We apply the Nehari manifold method in order to prove the main result.

On the effect of space dimension and potential on the multiplicity of positive and nodal solutions for Kirchhoff equations

The paper is concerned with the multiplicity of positive and nodal solutions for the following Kirchhoff equation arising in the study of string or membrane vibrations−(a+b∫RN|∇u|2dx)Δu+λV(x)u=|u|p−2u,inRN, where N≥3, a,b and λ are positive parameters, the potential V(x)∈C(RN,R+), and 2<p<min⁡{4,2⁎=2NN−2}. Because the complicated competition between nonlocal Kirchhoff term and sub-cubic nonlinearity, the standard nodal Nehari manifold method does not work. As we shall see, under some suitable assumptions on potential V(x), we prove that the above problem admits at least two positive solutions and two nodal solutions, where a novel constraint manifold and some analytical tricks are introduced to overcome this obstacle. An interesting finding is that the space dimension and potential have a strong influence on the number of positive or nodal solutions. Moreover, we also give the asymptotic behavior of solutions as λ→+∞ and b→0+. These results improve and generalize the previous results in the literature.

Existence of positive ground state solutions for the coupled Choquard system with potential

AbstractIn this paper, we study the following coupled Choquard system in : where and , in which denotes if and if . The function is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well‐studied case and , and the less‐studied case and . Moreover, it seems to be the first existence result for the case .

Convergence of least energy sign-changing solutions for logarithmic Schrödinger equations on locally finite graphs

In this paper, we study the following logarithmic Schrödinger equation −Δu+λa(x)u=ulogu2inVon a connected locally finite graph G=(V,E), where Δ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant λ0>0 such that for all λ≥λ0, the above problem admits a least energy sign-changing solution uλ. Moreover, as λ→+∞, we prove that the solution uλ converges to a least energy sign-changing solution of the following Dirichlet problem −Δu=ulogu2inΩ,u(x)=0on∂Ω,where Ω={x∈V:a(x)=0} is the potential well.

Existence of Positive Ground State Solutions for Fractional Choquard Systems in Subcritical and Critical Cases

We investigate a class of fractional linearly coupled Choquard systems. For the subcritical case and all critical cases, we prove the existence, nonexistence and symmetry of positive ground state solutions of systems, by using the Nehari manifold method, the Pohožaev identity and the Schwartz symmetrization rearrangements. In particular, we overcome the lack of compactness of the critical nonlinearities by using the behaviour of sufficiently small Nehari energy levels.

A study of extremal parameter for fractional singular Choquard problem

AbstractIn this work, we study the singular problem involving fractional Laplacian operator perturbed with a Choquard nonlinearity using the idea of constrained minimization based on Nehari manifold. Precisely, for some , we have proved the existence of two solutions when the parameter , adding to the existing works dealing with multiplicity of solutions when the parameter λ strictly lies below . We have given a variational characterization of the parametric value , which is an extremal value of the parameter λ involved in the problem up to which the Nehari manifold method can be applied successfully. The paper highlights a fine analysis via fibering maps even for to establish an existence of two different positive solutions for the underlying problem.

Existence and multiplicity of solutions for a class of Helmholtz systems

This paper establishes a dual variational framework for studying the existence and multiplicity of solutions to a class of Helmholtz systems. We first focus on a compact nonlinear potential and prove the existence and multiplicity of solutions via the symmetric Mountain Pass theorem. Next, we investigate the periodic nonlinear potential case and prove a nonvanishing theorem to recover the loss of compactness. We then demonstrate the existence of solutions using the Nehari manifold method and the multiplicity of solutions using the pseudoindex theory. Additionally, the regularity of solutions is given.

Existence and multiplicity of solutions for (p,q)$$ \left(p,q\right) $$‐Laplacian Kirchhoff‐type fractional differential equations with impulses

In this paper, we investigate the existence and multiplicity of solutions for a class of ‐Laplacian Kirchhoff‐type impulsive fractional differential equations. First, under a weaker condition than the Ambrosetti–Rabinowitz condition and the Miyagaki–Souto condition, the existence of an unbounded sequence of nontrivial solutions follows from the fountain theorem. Then, some new criteria are given to guarantee that the fractional differential equation has at least two nontrivial solutions using the Nehari manifold method combined with the fibering map.

Multiplicity and concentration of positive solutions for critical Choquard equations with concave perturbation

This paper focuses on three kinds of critical Choquard equations with concave perturbation λh(x)|u|q−2u, where λ>0, q∈(1,2) and h satisfies some mild conditions. For the upper critical and lower critical cases, by the Nehari manifold methods we obtain two positive solutions for λ>0 small enough, one of which is a ground state solution. For the doubly critical case, by minimization techniques and the mountain pass theorem, we derive two positive solutions for λ>0 small enough, one of which is a ground state positive solution. Moreover, the asymptotic behaviors of positive solutions as λ→0+ are verified.

The ground states of Hénon equations for [formula omitted]-Laplacian in [formula omitted] involving upper weighted critical exponents

In this paper, we investigate the existence of nontrivial ground state solutions of the Hénon equations with p-Laplacian in RN involving weighted upper critical exponents including Sobolev, Hardy–Sobolev and Hénon–Sobolev critical exponents. The Nehari manifold method and the mountain pass theorem are applied to get the nontrivial ground states for the problems under a subcritical weighted perturbation. Regularity and non-existence of the nontrivial solutions are also discussed.