Sign-changing Weight Research Articles

On the antimaximum principle for the discrete p-Laplacian with sign-changing weight

This work deals with the antimaximum principle for the discrete Neumann and Dirichlet problem−Δφp(Δu(k−1))=λm(k)|u(k)|p−2u(k)+h(k)in[1,n].We prove the existence of three real numbers 0 ≤ a < b < c such that, if λ ∈ ]a, b[, every solution u of this problem is strictly positive (maximum principle), if λ ∈ ]b, c[, every solution u of this problem is strictly negative (antimaximum principle) and if λ=b, the problem has no solution. Moreover these three real numbers are optimal.

Multiple solutions of critical singular degenerate elliptic system with concave-convex nonlinearities

This paper is devoted to study a class of critical singular degenerate elliptic system with concave-convex nonlinearities and sign-changing weight functions. The existence and multiplicity of nontrivial nonnegative solutions are obtained by the variational.

High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions

We prove that a class of superlinear indefinite problems with homogeneous Neumann boundary conditions admits an arbitrarily high number of positive solutions, provided that the parameters of the problem are adequately chosen. The sign-changing weight in front of the nonlinearity is taken to be piecewise constant, which allows us to perform a sharp phase-plane analysis, firstly to study the sets of points reached at the end of the regions where the weight is negative, and then to connect such sets through the flow in the positive part. Moreover, we study how the number of solutions depends on the amplitude of the region in which the weight is positive, using the latter as the main bifurcation parameter and constructing the corresponding global bifurcation diagrams.

Spectrum theory of second-order difference equations with indefinite weight

In this paper, we study the spectrum structure of second-order difference operators with sign-changing weight. We apply the Sylvester inertia theorem to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the j -th positive/negative eigenvalue changes its sign exactly j-1 times.

Three positive solutions for second-order periodic boundary value problems with sign-changing weight

In this paper, we study the global structure of positive solutions of periodic boundary value problems \t\t\t{−u″(t)+q(t)u(t)=λh(t)f(u(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π),\\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}$$\\textstyle\\begin{cases} -u''(t)+q(t)u(t)=\\lambda h(t)f(u(t)), \\quad t\\in (0,2\\pi ), \\\\ u(0)=u(2\\pi ), \\quad\\quad u'(0)=u'(2\\pi ), \\end{cases} $$\\end{document} where qin C([0,2pi ], [0, +infty )) with qnot equiv 0, fin C(mathbb{R},mathbb{R}), the weight hin C[0,2pi ] is a sign-changing function, λ is a parameter. We prove the existence of three positive solutions when h(t) has n positive humps separated by n+1 negative ones. The proof is based on the bifurcation method.

Multiplicity of solutions for a class of fractional p-Kirchhoff system with sign-changing weight functions

In this paper, we investigate the fractional p-Kirchhoff -type system: \t\t\t{M(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−Δ)psu=μg(x)|u|β−2u+aa+bh(x)|u|a−2u|v|b,in Ω,M(∫R2N|v(x)−v(y)|p|x−y|N+psdxdy)(−Δ)psv=σf(x)|v|β−2v+ba+bh(x)|v|b−2v|u|a,in Ω,u=v=0,in RN∖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} $$\\begin{aligned} \\textstyle\\begin{cases} M (\\int_{{ \\mathbb {R} }^{2N}}\\frac{\\vert u(x)-u(y) \\vert ^{p}}{\\vert x-y \\vert ^{N+ps}}\\,dx\\,dy )(- \\Delta )^{s}_{p}u=\\mu g(x)\\vert u \\vert ^{\\beta -2}u+\\frac{a}{a+b}h(x)\\vert u \\vert ^{a-2}u\\vert v \\vert ^{b},&\\mbox{in } \\Omega , \\\\ M (\\int_{{ \\mathbb {R} }^{2N}}\\frac{\\vert v(x)-v(y) \\vert ^{p}}{\\vert x-y \\vert ^{N+ps}}\\,dx\\,dy )(- \\Delta )^{s}_{p}v=\\sigma f(x)\\vert v \\vert ^{\\beta -2}v+\\frac{b}{a+b}h(x)\\vert v \\vert ^{b-2}v\\vert u \\vert ^{a},&\\mbox{in } \\Omega , \\\\ u=v=0,&\\mbox{in } { \\mathbb {R} }^{N}\\setminus \\Omega , \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where Omega subset mathbb{R}^{N} is a smooth bounded domain, (-Delta )^{s}_{p} is the fractional p-Laplacian operator with 0< s<1<p and ps< N . a>1, b>1 satisfy 2< a+b< p_{s}^{*}. 1<beta <p_{s}^{*}, p_{s}^{*}=frac{Np}{N-ps} is the fractional critical exponent. μ, σ are two real parameters. M(t)=k+lambda t^{tau }, k>0, λ, tau geq 0, tau =0 if and only if lambda =0. The weight functions g, f, h change sign in Ω and satisfy suitable conditions. By using the Nehari manifold method, it is proved that the system has at least two solutions provided that 2< a+b< pleq p(tau +1)<beta <p_{s}^{*} and (mu ,sigma ) belongs to a certain subset of mathbb {R} ^{2}. Also, by using the mountain pass theorem, we prove that there exist lambda _{1}geq lambda_{0} such that the system admits at least a nontrivial solution for lambda in (0,lambda_{0}) and no nontrivial solution for lambda >lambda_{1} under the assumptions mu =sigma =0 and p< a+b<min {p(tau +1),p_{s}^{*}}.

Properties of the first eigenvalue with sign-changing weight of the discrete p-Laplacian and applications

By establishing some results around the first eigenvalue λ1(m) for the following problem: -Δ(φp(Δu(k - 1)))= λm(k)φp(u(k)); k∈ [1; n]; u(0) = 0 = u(n + 1); where m ∈ M([1; n]) = {m : [1; n] → R /∃ k∈ [1; n]; m(k) &gt; 0} ; as the constant sign of the first eigenfunction with λ1(m); the simplicity of λ1(m); the strict monotonicity property with respect the weight and sign change of any eigenfunction with ( λ &gt; λ1(m)); we prove the existence and non-existence of solutions of the problem (1.1).

Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

Let $$\Omega \subset \mathbb {R}^{N}$$ ( $$N\ge 1$$ ) be a bounded and smooth domain and $$a:\Omega \rightarrow \mathbb {R}$$ be a sign-changing weight satisfying $$\int _{\Omega }a<0$$ . We prove the existence of a positive solution $$u_{q}$$ for the problem if $$q_{0}<q<1$$ , for some $$q_{0}=q_{0}(a)>0$$ . In doing so, we improve the existence result previously established in Kaufmann et al. (J Differ Equ 263:4481–4502, 2017). In addition, we provide the asymptotic behavior of $$u_{q}$$ as $$q\rightarrow 1^{-}$$ . When $$\Omega $$ is a ball and a is radial, we give some explicit conditions on q and a ensuring the existence of a positive solution of $$(P_{a,q})$$ . We also obtain some properties of the set of q’s such that $$(P_{a,q})$$ admits a solution which is positive on $$\overline{\Omega }$$ . Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method.

Infinitely many solutions for a class of quasilinear Schrödinger equations involving sign-changing weight functions

ABSTRACTUsing a change of variables and the constrained critical point theory, we first prove the existence and multiplicity of solutions for a class of quasilinear Schrödinger equations. Next, we consider a quasilinear equation related to the superfluid film in plasma physics with a sign-changing weight function. Using a new natural constraint, we establish the existence of infinitely many solutions for the equation.

Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Let T&gt;2 be an integer and T={1,2,…,T}. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem -Δ2u(j-1)+q(j)u(j)=λg(j)u(j), j∈T, u(0)=u(T), u(1)=u(T+1), and we determine the sign of the corresponding eigenfunctions, where λ is a parameter, q(j)≥0 and q(j)≢0 in T, and the weight function g changes its sign in T. As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem.

Multiple solutions for a Kirchhoff-type problem involving nonlocal fractional $p$-Laplacian and concave-convex nonlinearities

This paper examines a class of Kirchhoff nonlocal operators involving concave-convex nonlinearities and sign-changing weight functions. With the aid of the Nehari manifold, the existence of multiple nontrivial nonnegative solutions is obtained.

Multiplicity of positive solutions for fractional elliptic systems involving sign-changing weight

In this paper, we study the multiplicity results of positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of Morse theory and the Ljusternik-Schnirelmann category, we investigate how the coefficient h(x) of the critical nonlinearity affects the number of positive solutions of that problem and we get some results as regards the relationship between the number of positive solutions and the topology of the global maximum set of h.

Multiple solutions for the fractional differential equation with concave-convex nonlinearities and sign-changing weight functions

In this paper, by using the fibering map and the Nehari manifold, we prove the existence and multiple results of solutions for the following fractional differential equation: \t\t\t{DTαt(0Dtαu)=λh(t)|u|p−2u+b(t)|u|q−2u,t∈[0,T],u(0)=u(T)=0,\\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} $$\\begin{aligned} \\textstyle\\begin{cases} {_{t}}D{_{T}^{\\alpha}}({_{0}}D{_{t}^{\\alpha}}u)=\\lambda h(t) \\vert u \\vert ^{p-2}u+b(t) \\vert u \\vert ^{q-2}u,\\quad t\\in[0,T],\\\\ u(0)=u(T)=0, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where alphain(frac{1}{2},1), 0< p<2, q>2, lambda>0 and h(t),b(t) are sign-changing continuous functions.

Multiple positive solutions for a fractional elliptic systems involving sign-changing weight

We study multiplicity results for positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $h$ of the critical nonlinearity affects the number of positive solutions to this problem and get a relationship between the number of positive solutions and the topology of the global maximum set of $h$.

Large solutions for a semilinear elliptic problem with sign-changing weights

This paper focuses on an elliptic boundary blow up problem with sign-changing weights△uϵ=(a+−ϵa−)g(uϵ), x∈Ω,uϵ|∂Ω=+∞, where Ω⊂Rn(n≥1) is a bounded smooth domain and ϵ>0 is a real parameter. We verify that there exists ϵ⁎>0 such that the problem has the minimal positive large solution for any ϵ∈(0,ϵ⁎), while no positive large solutions for any ϵ>ϵ⁎. Then, by applying the local bifurcation theory, we analyze the structure of the branch produced by the positive large solutions.

On a two-component Bose–Einstein condensate with steep potential wells

In this paper, we study the following two-component systems of nonlinear Schrodinger equations $$\begin{aligned} \left\{ \begin{array}{ll} \Delta u-(\lambda a(x)+a_0(x))u+\mu _1u^3+\beta v^2u=0&{}\quad \text {in }\mathbb \Delta v-(\lambda b(x)+b_0(x))v+\mu _2v^3+\beta u^2v=0&{}\quad \text {in }\mathbb u,v\in H^1(\mathbb {R}^3), u,v>0&{}\quad \text {in }\mathbb {R}^3, \end{array}\right. \end{aligned}$$ where \(\lambda ,\mu _1,\mu _2>0\) and \(\beta <0\) are parameters; \(a(x), b(x)\ge 0\) are steep potentials and \(a_0(x),b_0(x)\) are sign-changing weight functions; a(x), b(x), \(a_0(x)\) and \(b_0(x)\) are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with \(\lambda \) sufficiently large. The concentration behaviors of solutions as both \(\lambda \rightarrow +\infty \) and \(\beta \rightarrow -\infty \) are also considered. In particular, the phenomenon of phase separations is observed in the whole space \(\mathbb {R}^3\). In the Hartree–Fock theory, this provides a theoretical enlightenment of phase separation in \(\mathbb {R}^3\) for the 2-mixtures of Bose–Einstein condensates.

Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree

We study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

Comment on “Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications”

In the recent paper W. Shen and T. He and G. Dai and X. Han established unilateral global bifurcation result for a class of nonlinear fourth-order eigenvalue problems. They show the existence of two families of unbounded continua of nontrivial solutions of these problems bifurcating from the points and intervals of the line trivial solutions, corresponding to the positive or negative eigenvalues of the linear problem. As applications of this result, these authors study the existence of nodal solutions for a class of nonlinear fourth-order eigenvalue problems with sign-changing weight. Moreover, they also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight. In the present comment, we show that these papers of above authors contain serious errors and, therefore, unfortunately, the results of these works are not true. Note also that the authors used the results of the recent work by G. Dai which also contain gaps.

Multiple Solutions to the Problem of Kirchhoff Type Involving the Critical Caffareli-Kohn-Niremberg Exponent, Concave Term and Sign-Changing Weights

In this paper, we establish the existence of at least four distinct solutions to an Kirchhoff type problems involving the critical Caffareli-Kohn-Niremberg exponent, concave term and sign-changing weights, by using the Nehari manifold and mountain pass theorem.

Multiplicity results for a fractional Kirchhoff equation involving sign-changing weight function

In this paper, we prove the existence and multiplicity of solutions for a fractional Kirchhoff equation involving a sign-changing weight function which generalizes the corresponding result of Tsung-fang Wu (Rocky Mt. J. Math. 39:995-1011, 2009). Our main results are based on the method of a Nehari manifold.