Nehari Manifold Method Research Articles

Ground-state nodal solutions for superlinear perturbations of the Robin eigenvalue problem

We consider a perturbed version of the Robin eigenvalue problem for the p-Laplacian. The perturbation is (p - 1)-superlinear. Using the Nehari manifold method, we show that for all parameters lambda < {hat{lambda }}_1 (= the principal eigenvalue of the differential operator), there exists a ground-state nodal solution of the problem.

Existence and asymptotic behaviour of positive ground state solution for critical Schrödinger-Bopp-Podolsky system

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a class of critical Schrödinger-Bopp-Podolsky system. By virtue of the Nehari manifold and variational methods, we study the existence, nonexistence and asymptotic behavior of ground state solutions for this problem.&lt;/p&gt;&lt;/abstract&gt;

Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: \t\t\t−Δu+V(x)u=[|x|−μ∗|u|p]|u|p−2u,x∈RN,\\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} $$\\begin{aligned} -\\Delta u+V(x)u=\\bigl[ \\vert x \\vert ^{-\\mu }\\ast \\vert u \\vert ^{p}\\bigr] \\vert u \\vert ^{p-2}u,\\quad x \\in \\mathbb{R}^{N}, \\end{aligned}$$ \\end{document} where Ngeq 3, 0<mu <N, frac{2N-mu }{N}leq p<frac{2N-mu }{N-2}, ∗ represents the convolution between two functions. We assume that the potential function V(x) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

Schrödinger p⋅–Laplace equations in RN involving indefinite weights and critical growth

We study a class of critical Schrödinger p⋅–Laplace equations in RN, with reaction terms of the concave–convex type and involving indefinite weights. The class of potentials used in this study is different from that in most existing studies on Schrödinger equations in RN. We establish a concentration-compactness principle for weighted Sobolev spaces with variable exponents involving the potentials. By employing this concentration-compactness principle and the Nehari manifold method, we obtain existence and multiplicity results for the solution to our problem.

Multiple solutions for superlinear double phase Neumann problems

We study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

The ground state solutions of Hénon equation with upper weighted critical exponents

In the present paper we study the existence of ground state solutions for Hénon equations involving with single or multiple upper weighted critical exponents with general perturbational term, where the upper weighted critical exponents include upper Hardy-Sobolev, Sobolev or Hénon-Sobolev critical exponents for the embedding from Hr1(RN) into Lq(RN;|x|α) with α>−2. The Nehari manifold method and the mountain pass theorem are applied to obtain the ground state solutions with different assumptions on general perturbational term. The existence of ground state solutions is closely related to the sign of the parameters in the equations.

Positive solutions for the fractional Schrödinger equations with logarithmic and critical non‐linearities

In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non-linearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.

Concentration phenomena for fractional magnetic NLS equations

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Ground state and nodal solutions for critical Schrödinger–Kirchhoff-type Laplacian problems

In this paper, we are interested in the existence of ground state nodal solutions for the following Schrodinger–Kirchhoff-type Laplacian problems: $$\begin{aligned} -M\bigg (\int _{{\mathbb {R}}^{3}}|\nabla u|^{2}\mathrm{{d}}x\bigg )\Delta u+V(x)u=|u|^{4}u+ k f(u),\;x\in {\mathbb {R}}^{3}, \end{aligned}$$ where $$M(t)=a+bt^\gamma $$ with $$0<\gamma <2$$ , $$a,b>0$$ and the nonlinear function $$f\in C({\mathbb {R}},{\mathbb {R}})$$ . By the nodal Nehari manifold method, for each $$b>0$$ , we obtain a least energy nodal solution $$u_{b}$$ and a ground state solution $$v_b$$ of this problems when $$k\gg 1$$ . Our results improve and extend the known results of the usual case $$\gamma =1$$ in the sense that a more wider range of $$\gamma $$ is covered.

Existence and multiplicity results for critical and subcritical $p$-fractional elliptic equations via Nehari manifold method

Existence and multiplicity results for critical and subcritical $p$-fractional elliptic equations via Nehari manifold method

Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity

In this paper, we consider the following fractional Kirchhoff problem with strong singularity: {(1+b∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy)(−Δ)su+V(x)u=f(x)u−γ,x∈R3,u>0,x∈R3,\\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}$$ \\textstyle\\begin{cases} (1+ b\\int _{\\mathbb{R}^{3}}\\int _{\\mathbb{R}^{3}} \\frac{ \\vert u(x)-u(y) \\vert ^{2}}{ \\vert x-y \\vert ^{3+2s}}\\,\\mathrm{d}x \\,\\mathrm{d}y )(-\\Delta )^{s} u+V(x)u = f(x)u^{-\\gamma }, & x \\in \\mathbb{R}^{3}, \\\\ u>0,& x\\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where (-Delta )^{s} is the fractional Laplacian with 0< s<1, b>0 is a constant, and gamma >1. Since gamma >1, the energy functional is not well defined on the work space, which is quite different with the situation of 0<gamma <1 and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution u_{b} by using variational methods and the Nehari manifold method. We also give a convergence property of u_{b} as brightarrow 0, where b is regarded as a positive parameter.

Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity

This paper focuses on the study of ground states and multiple solutions for the following non-autonomous Choquard-Pekar equation:{−Δu+V(x)u=(W⁎F(x,u))f(x,u),x∈RN(N≥2),u∈H1(RN), where V∈C(RN,R). We consider first the case V changes sign which turns the problem into a indefinite case, and obtain the existence of nontrivial solution and infinitely many distinct pairs of solutions under a local super-linear condition assumed on the nonlinearity. For the case V is 1-periodic and positive, ground state solution and infinitely many solutions are established further by using the generalized Nehari manifold method. We finally give some non-existence criteria via a generalized Pohožaev identity established for the general potentials V and W.

Positive solutions of the prescribed mean curvature equation with exponential critical growth

In this paper, we are concerned with the problem -div∇u1+|∇u|2=f(u)inΩ,u=0on∂Ω,\\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}$$\\begin{aligned} -\\text{ div } \\left( \\displaystyle \\frac{\\nabla u}{\\sqrt{1+|\\nabla u|^2}}\\right) = f(u) \\ \\text{ in } \\ \\Omega , \\ \\ u=0 \\ \\text{ on } \\ \\ \\partial \\Omega , \\end{aligned}$$\\end{document}where Omega subset {mathbb {R}}^{2} is a bounded smooth domain and f:{mathbb {R}}rightarrow {mathbb {R}} is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.

Least energy sign-changing solutions for a class of Schrödinger–Poisson system on bounded domains

This paper is concerned with the Schrödinger–Poisson system −Δu + ϕu = λu + μ|u|2u and −Δϕ = u2 setting on a bounded domain Ω⊂R3 with smooth boundary and λ,μ∈R being parameters. By using variational techniques in combination with the nodal Nehari manifold method, we show the existence of μ̄&amp;gt;0 such that for all (λ,μ)∈(−∞,λ1)×(μ̄,+∞), the above system has one least energy sign-changing solution, where λ1 &amp;gt; 0 is the first eigenvalue of −Δ,H01(Ω). The results of this paper are complementary to those in Alves and Souto [Z. Angew. Math. Phys. 65, 1153–1166 (2014)].

Positive ground state solutions for Choquard equations with lower critical exponent and steep well potential

In this paper, we investigate the following non-autonomous Choquard equation −Δu+λV(x)−μu=Iα∗|u|2∗α|u|2∗α−2uinRN,where N≥3, λ>0, μ∈R, Iα is a Riesz potential of order α∈(0,N), 2∗α=N+αN, the function V∈C(RN,R) is nonnegative and has a potential well Ω≔intV−1(0) such that −Δ possesses a sequence of Dirichlet eigenvalues 0<μ1<μ2<⋯<μn→+∞ on Ω. Assume μ<μ1, by the Nehari manifold method we prove the existence and concentration of positive ground state solutions for λ large enough.

Multiple solutions for a coupled Kirchhoff system with fractional p-Laplacian and sign-changing weight functions

In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains: By using the Nehari manifold method, together with Ekeland's variational principle, we show that there exist two distinct solutions under suitable conditions on weight functions and h. Our results extend and generalize the main results in Xiang et al. [Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity. 2016;29:3186–3205] in Nonlinearity 2016, in which are constants.

Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold

In this paper we study quasilinear elliptic equations driven by the so-called double phase operator and with a nonlinear boundary condition. Due to the lack of regularity, we prove the existence of multiple solutions by applying the Nehari manifold method along with truncation and comparison techniques and critical point theory. In addition, we can also determine the sign of the solutions (one positive, one negative, one nodal). Moreover, as a result of independent interest, we prove for a general class of such problems the boundedness of weak solutions.

Ground state solutions for a class of nonlocal regional Schrödinger equation with nonperiodic potentials

In this article, we deal with the nonlinear Schrödinger equation with nonlocal regional diffusion where 0 &lt; α &lt; 1, n ≥ 2, and is a continuous function. The operator is a variational version of the nonlocal regional Laplacian defined as where be a positive function. Considering that ρ, V, and f(· , t) are periodic or asymptotically periodic at infinity, we prove the existence of ground state solution of (1) by using Nehari manifold and comparison method. Furthermore, in the periodic case, by combining deformation‐type arguments and Lusternik–Schnirelmann theory, we prove that problem (1) admits infinitely many geometrically distinct solutions.

Nontrivial solutions for a quasilinear elliptic system with weight functions

In this paper, we consider nontrivial solutions of a quasilinear elliptic system with the weight functions $h(x)$ and $f_i(x)$ $(i = 1, 2)$ by applying the Nehari manifold method along with the fibrering maps and the minimization method. We analyze the effect of the parameters and weight functions on the existence and multiplicity of nontrivial solutions for the quasilinear elliptic system. When $(\lambda_1 f_1)^+ = (\lambda_2 f_2)^+ \equiv 0$, we prove that the system has at least one nontrivial nonnegative solution; and when $(\lambda_1 f_1)^+ \not \equiv 0$ or $(\lambda_2 f_2)^+ \not \equiv 0$, we show that the system has at least two nontrivial nonnegative solutions for any $(\lambda_1, \lambda_2)$ belonging to a certain subset of $\mathbb{R}^2$. These results are novel even for the corresponding scalar quasilinear elliptic equations and semilinear elliptic systems, which improve and extend the existing results in the literature.

Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential in ℝ N ∗

In this paper, we show the existence of solutions for an indefinite fractional Schrödinger equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents and steep potential. By using the decomposition of the Nehari manifold and variational method, we obtain the existence results of nontrivial solutions to the equation under suitable conditions.