Nehari Manifold 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.

Indefinite perturbations of an unbalanced growth eigenvalue problem

We consider indefinite perturbations of a double-phase eigenvalue problem. The perturbation is sublinear or superlinear, and it is in general sign-changing. Using the Nehari manifold, we prove the existence of two constant sign solutions for both cases (sublinear and superlinear), when the parameter \lambda\in (0,\hat{\lambda}^{\alpha}_{1}) with \hat{\lambda}^{\alpha}_{1}>0 being the principal eigenvalue of Dirichlet weighted p -Laplace operator -\Delta^{\alpha}_{p} .

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 .

Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs

In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

Fractional Sobolev space: Study of Kirchhoff-Schrödinger systems with singular nonlinearity

This study extensively investigates a specific category of Kirchhoff-Schrödinger systems in fractional Sobolev space with Dirichlet boundary conditions. The main focus is on exploring the existence and multiplicity of non-negative solutions. The non-linearity of the problem generally exhibits singularity. By employing minimization arguments involving the Nehari manifold and a variational approach, we establish the existence and multiplicity of positive solutions for our problem with respect to the parameters \(\eta\) and \(\zeta\) in suitable fractional Sobolev spaces. Our key findings are novel and contribute significantly to the literature on coupled systems of Kirchhoff-Schrödinger system with Dirichlet boundary conditions.

The asymptotic behavior of constant sign and nodal solutions of [formula omitted]-Laplacian problems as [formula omitted] goes to 1

In this paper we study the asymptotic behavior of solutions to the (p,q)-equation −Δpu−Δqu=f(x,u)inΩ,u=0on∂Ω,as p→1+, where N≥2, 1<p<q<1∗≔N/(N−1) and f is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1,q)-Laplace problem given by −div∇u|∇u|−Δqu=f(x,u)inΩ,u=0on∂Ω,has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space W01,q(Ω) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(Ω) of all functions of bounded variation. As far as we know this is the first work dealing with (1,q)-Laplace problems even in the direction of constant sign solutions.

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.

Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients

We consider the following Neumann boundary problem with a nonlocal Kirchhoff term where is a bounded smooth domain and and are sign‐changing coefficients which play important impact on the study. By using variational method and dividing corresponding Nehari manifold into two disjoint subsets, we study the existence of two positive solutions. To show the multiplicity of positive solutions, we find a set whose elements can be mapped to two subsets of Nehari manifold, which is also the key to proving strong convergence for minimizing sequences. Finally, the asymptotic behavior of solutions is studied as parameter tends to 0. Different from Dirichlet boundary problem, we show that the two solutions of Neumann problem tend to two different positive constants, respectively.

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

Multiple solutions for a Bibarmonic elliptic equation with critical Choquard type nonlinearity

This article concerns the following Bibarmonic elliptic equation with critical growth where Ω ⊂ ℝ N is a smooth bounded domain , g(x) is a positive continuous function on is a real parameter, is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. By combining the Nehari manifold and the variational methods, we prove the existence of k nontrivial solutions for this equation. Our results extend the results of Rani and Goyal in Topological Methods in Nonlinear Analysis 59 (2022), 221–260.

Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions

AbstractIn this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.

The Nehari Manifold for a Class of Singular $\psi$-Riemann-Liouville Fractional with $p$-Laplacian Operator Differential Equations

The Nehari Manifold for a Class of Singular $\psi$-Riemann-Liouville Fractional with $p$-Laplacian Operator Differential Equations

Existence of Weak Solutions for the Class of Singular Two-Phase Problems with a ψ-Hilfer Fractional Operator and Variable Exponents

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results.

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.

Multiple positive solutions for a critical fractional p-Laplacian system with concave nonlinearities

The aim of this paper is to study the following nonlinear fractional p-Laplacian system with critical exponents: { ( − Δ ) p s u + | u | p − 2 u = λg ( x ) | u | q − 2 u + α α + β f ( x ) | u | α − 2 u | v | β in Ω , ( − Δ ) p s v + | v | p − 2 v = μh ( x ) | v | q − 2 v + β α + β f ( x ) | u | α | v | β − 2 v in Ω , u = v = 0 in R N ∖Ω , where Ω is a smooth bounded set in R N , 0<s<1, 0 $ ]]> λ , μ > 0 are two parameters, 1 < q < p < p s ∗ , N>ps, 1 $ ]]> α , β > 1 satisfy α + β = p s ∗ with p s ∗ = np n − ps is the fractional Sobolev critical exponent and ( − Δ ) p s is the fractional p-Laplacian operator. Using the Nehari manifold and Ljusternik–Schnirelmann category, we study the topology of the global maximum set Θ of f ( x ) , and show that the system has at least at least ca t Θ δ ( Θ ) + 1 distinct positive solutions.

Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

Abstract The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x &lt; + ∞ , $$\underset{u\in {H}^{1}\left({\mathbb{R}}^{2}\right),\underset{{\mathbb{R}}^{2}}{\int }\left(\vert \nabla u{\vert }^{2}+V\left(x\right){u}^{2}\right)\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\frac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x{&lt; }+\infty ,$$ where 0 &lt; β &lt; 2 $0{&lt; }\beta {&lt; }2$ , V ( x ) ≥ 0 $V\left(x\right)\ge 0$ and may vanish on an open set in R 2 ${\mathbb{R}}^{2}$ . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 ${\mathbb{R}}^{2}$ : (0.2) − Δ u + γ u = f ( u ) | x | β , $$-{\Delta}u+\gamma u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }},$$ where the nonlinearity f $f$ has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in R n ${\mathbb{R}}^{n}$ and n $n$ -Laplace equations,” Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 ${\mathbb{R}}^{2}$ : (0.3) − Δ u + V ( x ) u = f ( u ) | x | β . $$-{\Delta}u+V\left(x\right)u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }}.$$

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.

Existence of the Solution for a Double Phase System with Convex Nonlinearities

In this paper, we study the following double phase system which contains the convex nonlinearities. By the use of the Nehari manifold, the existence of one nontrivial solution which has nonnegative energy is obtained.

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 &lt; q &lt; p &lt; 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 .