Sign-changing Nonlinearity Research Articles

Hardy–Sobolev equation with negative power and sign-changing nonlinearity on closed manifolds

Hardy–Sobolev equation with negative power and sign-changing nonlinearity on closed manifolds

Existence of nontrivial solutions for a fractional p&q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\&q$\\end{document}-Laplacian equation with sandwich-type and sign-changing nonlinearities

In this paper, we deal with the following fractional p&q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\&q$\\end{document}-Laplacian problem: {(−Δ)psu+(−Δ)qsu=λa(x)|u|θ−2u+μb(x)|u|r−2uinΩ,u(x)=0inRN∖Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left \\{ \ extstyle\\begin{array}{l@{\\quad }l} (-\\Delta )_{p}^{s}u +(-\\Delta )_{q}^{s}u =\\lambda a(x)|u|^{\ heta -2}u+ \\mu b(x)|u|^{r-2}u&\ ext{in}\\;\\ \\Omega , \\\\ u(x)=0 &\ ext{in}\\;\\ \\mathbb{R}^{N}\\setminus \\Omega , \\end{array}\\displaystyle \\right . $$\\end{document} where Ω⊂RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset \\mathbb{R}^{N}$\\end{document} is a bounded domain with smooth boundary, s∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s\\in (0,1)$\\end{document}, (−Δ)ms\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(-\\Delta )_{m}^{s}$\\end{document}(m∈{p,q})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m\\in \\{p,q\\})$\\end{document} is the fractional m-Laplacian operator, p,q,r,θ∈(1,ps∗]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p,q,r,\ heta \\in (1,p_{s}^{*}]$\\end{document}, ps∗=NpN−sp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p_{s}^{*}=\\frac{Np}{N-sp}$\\end{document}, λ,μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda , \\mu >0$\\end{document}, and the weights a(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a(x)$\\end{document} and b(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$b(x)$\\end{document} are possibly sign changing. Using the concentration compactness principle for fractional Sobolev spaces and the Ekeland variational principle, we prove that the problem admits a nonnegative solution for the critical case r=ps∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r=p_{s}^{*}$\\end{document}. Moreover, for the subcritical case r<ps∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r< p_{s}^{*}$\\end{document}, we obtain two existence results by applying the Ekeland variational principle and the mountain pass theorem.

On Dancer’s conjecture for stable solutions with sign-changing nonlinearity

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in R n \mathbb {R}^{n} with sign-changing nonlinearity f f . Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of f f at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer.

Existence and multiplicity of solutions for boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel’skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively.

On a class of singular double phase problems with nonnegative weights whose sum can be zero

We study the existence, nonexistence and multiplicity of positive solutions for a class of one-dimensional double phase problems with nonnegative weights whose sum can be zero, singular nonlinearities and nonlinear boundary conditions. Such a weight condition is interesting in that it weakens the regularities of solutions, making it challenging to construct solution operators corresponding to the problems. We use a fixed point theorem to establish the existence and multiplicity of positive solutions according to the behaviors of the nonlinearity near 0 and ∞. In particular, we find intervals that guarantee at least three positive solutions for the problems with positive nonlinearities and at least two positive solutions for the problems with sign-changing nonlinearities.

Positive Solutions for Periodic Boundary Value Problems of Fractional Differential Equations with Sign-Changing Nonlinearity and Green’s Function

In this paper, a class of nonlinear fractional differential equations with periodic boundary condition is investigated. Although the nonlinearity of the equation and the Green’s function are sign-changing, the results of the existence and nonexistence of positive solutions are obtained by using the Schaefer’s fixed-point theorem. Finally, two examples are given to illustrate the main results.

Fractional p-Laplacian elliptic problems with sign changing nonlinearities via the nonlinear Rayleigh quotient

It is established existence and multiplicity of solutions to the fractional p-Laplacian problem in the whole space RN. More precisely, we consider the nonlocal elliptic problem with sign changing nonlinearities in the following form:{(−Δ)psu+V(x)|u|p−2u=λf(x)|u|q−2u+g(x)|u|r−2uinRN,u∈Ws,p(RN), where λ∈(γ⁎,0)∪(0,λ⁎),γ⁎<0, λ⁎>0 and N>ps with s∈(0,1) fixed. Furthermore, we assume that 1<q<p<r<ps⁎=Np/(N−ps). The potential V is a continuous function which is bounded from below by a positive constant. The main objective here is to consider nonlinearities f and g that can be sign changing functions. In this case, by using the nonlinear Rayleigh quotient, we prove that our main problem has at least two nontrivial solutions for each λ∈(γ⁎,0)∪(0,λ⁎). More specifically, the numbers λ⁎>0 and γ⁎<0 are sharp in order to consider the Nehari method, that is, the number λ⁎ is the largest positive number such that the Nehari method can be applied for each λ∈(0,λ⁎). The same assertion is verified for γ⁎, that is, the number γ⁎<0 is the smallest negative number such that the Nehari method can be employed for each λ∈(γ⁎,0).

The existence of positive solutions for high order fractional differential equations with sign changing nonlinearity and parameters

&lt;abstract&gt;&lt;p&gt;By constructing an auxiliary boundary value problem, the difficulty caused by sign changing nonlinearity terms is overcome by means of the linear superposition principle. Using the Guo-Krasnosel'skii fixed point theorem, the results of the existence of positive solutions for boundary value problems of high order fractional differential equation are obtained in different parameter intervals under a more relaxed condition compared with the existing literature. As an application, we give two examples to illustrate our results.&lt;/p&gt;&lt;/abstract&gt;

Existence of solutions for a delay singular high order fractional boundary value problem with sign-changing nonlinearity

This paper consider the existence of at least one positive solution of a Riemann-Liouville fractional delay singular boundary value problem with sign-changing nonlinerty. To establish sufficient conditions we use the Guo-Krasnosel?skii fixed point theorem.

Positive solutions for nonlinear singular problems with sign-changing nonlinearities

Positive solutions for nonlinear singular problems with sign-changing nonlinearities

Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity

Abstract In this article, we will prove the existence of infinitely many solutions for a class of quasilinear Schrödinger equations without assuming the 4-superlinear at infinity on the nonlinearity. We achieve our goal by using the Fountain theorem.

On an anisotropic double phase problem with singular and sign changing nonlinearity

This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case are new also in the classical p-Laplacian case.

Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearities

We show a linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schrödinger equation $$\begin{aligned} -\Delta u + V(x)u + \frac{a}{r^2} u = f(u) - \lambda g(u), \quad x = (y,z) \in {\mathbb {R}}^K \times {\mathbb {R}}^{N-K}, \ r = |y|, \end{aligned}$$ where $$\begin{aligned} 0 \not \in \sigma \left( -\Delta + \frac{a}{r^2} + V(x) \right) . \end{aligned}$$ As a consequence we obtain also the existence of solutions to the nonlinear curl-curl problem.

Multiple solutions for sublinear Schrödinger-Poisson equations

We study the multiplicity result to the nonlinear equations with sign-changing potential here Ω is a bounded domain, λ> 0, g is a sign-changing nonlinearity and is a sublinear growth. Under appropriate assumptions for a(x), the multiplicity result is proved for λ >0 small enough.

Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity

In this work we consider the following class of elliptic problems $$- \Delta_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H^1_A (\mathbb{R}^N)$, with $2<q<p<2^*= \frac{2N}{N-2}$, $a(x)$ and $b(x)$ are functions that can change signal and satisfy some additional conditions; $u \in H^1_A(\mathbb{R}^N)$ and $A:\mathbb{R}^N \rightarrow \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinite solutions to the problem in question, varying the assumptions about the weight functions.

Existence of positive solutions to the fractional Kirchhoff-type problems involving steep potential well

The aim of this paper is to study the existence of positive solutions to the fractional Kirchhoff-type problems involving steep potential well and sign-changing nonlinearity . Applying truncation technique and mountain pass theorem, we investigate the existence of positive solutions under suitable assumptions when 2<p<4.

Existence of solutions for a Dirac equation with sign-changing nonlinearity

In this paper, we consider a class of periodically nonlinear Dirac equations. Under certain assumptions, we obtain that there exists at least one non-trivial solution for the super-linear Dirac equation with sign-changing nonlinearity via the variational method.

Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form -Δu-k2u=Q(x)|u|p-2u,u∈W2,pRN\\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}$$\\begin{aligned} - \\Delta u - k^{2}u = Q(x)|u|^{p-2}u, \\quad u \\in W^{2,p}\\left( {\\mathbb {R}}^{N}\\right) \\end{aligned}$$\\end{document}with k>0,N ge 3, p in left[ left. frac{2(N+1)}{N-1},frac{2N}{N-2}right) right. and Q in L^{infty }({mathbb {R}}^{N}). Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

Positive radial solutions for a noncooperative resonant nuclear reactor model with sign-changing nonlinearities

This paper is concerned with the existence of positive radial solutions of the following resonant elliptic system: \t\t\t{−Δu=uv+f(|x|,u),0<R1<|x|<R2,x∈RN,−Δv=cg(u)−dv,0<R1<|x|<R2,x∈RN,∂u∂n=0=∂v∂n,|x|=R1,|x|=R2,\\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}$$ \\textstyle\\begin{cases} -\\Delta u=uv+f( \\vert x \\vert ,u), & 0< R_{1}< \\vert x \\vert < R_{2}, x\\in \\mathbb{R}^{N}, \\\\ -\\Delta v=cg(u)-dv, & 0< R_{1}< \\vert x \\vert < R_{2}, x\\in \\mathbb{R}^{N}, \\\\ \\frac{\\partial u}{\\partial \\textbf{n}}=0= \\frac{\\partial v}{\\partial \\textbf{n}},& \\vert x \\vert =R_{1}, \\vert x \\vert =R_{2}, \\end{cases} $$\\end{document} where mathbb{R}^{N} (Ngeq 1) is the usual Euclidean space, n indicates the outward unit normal vector, fin C([R_{1},R_{2}]times [0,infty ),mathbb{R}), gin C([0,infty ),[0,infty )), and c and d are positive constants. By employing the classical fixed point theory we establish several novel existence theorems. Our main findings enrich and complement those available in the literature.

Nontrivial Solutions for the 2 n th Lidstone Boundary Value Problem

In this paper, we study the existence of nontrivial solutions for the 2 n th Lidstone boundary value problem with a sign-changing nonlinearity. Under some conditions involving the eigenvalues of a linear operator, we use the topological degree theory to obtain our main results.