Sign-changing Solutions Research Articles

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

New results concerning a singular biharmonic equations with p-Laplacian and Hardy potential

This paper deals with a singular biharmonic equations with p-Laplacian and Hardy potential. Using Mountain Pass Theorem and Fountain Theorem with Cerami condition, we obtain the existence and multiplicity of sign-changing high-energy solutions under some suitable conditions on the nonlinear term f ( x , u ) . In addition, by applying an abstract critical point theorem of Kajikiya in [A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370], a sequence of sign-changing solutions converging to zero for the biharmonic equations with sublinear nonlinearities is also obtained.

Infinitely many sign-changing solutions for the quasilinear Schrödinger-Poisson system

Infinitely many sign-changing solutions for the quasilinear Schrödinger-Poisson system

Sequences of nodal solutions for critical double phase problems with variable exponents

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} and in the Musielak–Orlicz Sobolev space.

Sign-changing bubble tower solutions for a Paneitz-type problem

This paper is concerned with the following biharmonic problem {Δ2u=|u|8N−4uin Ω∖B(ξ0,ε)―,u=Δu=0on ∂(Ω∖B(ξ0,ε)―), where Ω is an open bounded domain in RN , N⩾5 , and B(ξ0,ε) is a ball centered at ξ 0 with radius ɛ, ɛ is a small positive parameter. We obtain the existence of solutions for problem (0.1), which is an arbitrary large number of sign-changing solutions whose profile is a superposition of bubbles with alternate sign which concentrate at the centre of the hole.

Multiplicity and limit of solutions for logarithmic Schrödinger equations on graphs

Let Ω be a finite connected subset of a locally finite graph G = (V, E) with the vertex set V and the edge set E. We investigate the logarithmic Schrödinger equation on Ω with the nonlinear term |u|p−2u log u2. For p > 2, through two different approaches which are the Brouwer degree theory and mountain-pass theorem, we obtain the existence of ground state solutions. We also apply the Brouwer degree theory together with the constraint variational method to prove that the equation admits a sign-changing solution which implies the multiplicity of solutions to the equation. Finally, we illustrate that as p → 2, up to a subsequence, the solutions for p > 2 shall converge to a non-trivial solution of the equation with p = 2.

Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator

Abstract We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator − u ′ 1 − u ′ 2 ′ = λ a ( x ) f ( u , u ′ ) , x ∈ R , $-{\left(\frac{{u}^{\prime }}{\sqrt{1-{u}^{\prime 2}}}\right)}^{\prime }=\lambda a\left(x\right)f\left(u,{u}^{\prime }\right), x\in \mathbb{R},$ where λ > 0 is a parameter, a ∈ C ( R , R ) $a\in C\left(\mathbb{R},\mathbb{R}\right)$ is a 2π-periodic sign-changing function with ∫ 0 2 π a ( x ) d x < 0 ${\int }_{0}^{2\pi }a\left(x\right)\mathrm{d}x{< }0$ , f ∈ C ( R × R , R ) $f\in C\left(\mathbb{R}{\times}\mathbb{R},\mathbb{R}\right)$ satisfies a generalized regular-oscillation condition. Moreover, for the special case that f does not contain derivative term, we also investigate the global structure of 2π-periodic odd/even sign-changing solutions set under some parity conditions. The proof of our main results are based upon bifurcation techniques.

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Holder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

The Sign-Changing Solution for Fractional (p,q)-Laplacian Problems Involving Supercritical Exponent

In this article, we consider the following fractional (p,q)-Laplacian problem (−Δ)ps1u+(−Δ)qs2u+V(x)(|u|p−2u+|u|q−2u)=f(u)+λ|u|r−2u, where x∈RN, (−Δ)ps1 is the fractional p-Laplacian operator ((−Δ)qs2 is similar), 0<s1<s2<1<p<q<Ns2, qs2*=NqN−s2q, r≥qs2*, f is a C1 real function and V is a coercive function. By using variational methods, we prove that the above problem admits a sign-changing solution if λ>0 is small.

An upper bound for the least energy of a sign-changing solution to a zero mass problem

Abstract We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c 0 if N = 5, 6 and smaller than 10c 0 if N ≥ 7, where c 0 is the ground state energy.

Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities

Least Energy Sign-Changing Solution for N-Kirchhoff Problems with Logarithmic and Exponential Nonlinearities

A fractional Hopf Lemma for sign-changing solutions

In this paper we prove some results on the boundary behavior of solutions to fractional elliptic problems. Firstly, we establish a Hopf Lemma for solutions to some integro-differential equations. The main novelty of our result is that we do not assume any global condition on the sign of the solutions. Secondly, we show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary. We provide concrete examples to show that the results obtained are sharp.

Equivariant Solutions to the Optimal Partition Problem for the Prescribed Q-Curvature Equation

We study the optimal partition problem for the prescribed constant Q-curvature equation induced by the higher-order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed Q-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant Q-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of GJMS operators on Ricci solitons, compute the Q-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with a prescribed number of nodal domains on the Koiso–Cao Ricci soliton.

Positivity Expansion to Signed Solution for Doubly Nonlinear Parabolic Equations

We investigate doubly nonlinear parabolic equation with sign changing solutions. We established the expansion of positivity to the sign changing solution within a parabolic domain which is the key elements to achieve the regularity results. Jagannath University Journal of Science, Volume 10, Number I, Jun. 2023, pp. 64−74

Sign-changing solutions for a weighted Kirchhoff problem with exponential growth non-linearity

In this work, we establish the existence of solutions that change sign at low energy for a non-local weighted Kirchhoff problem in the set }2 $ ]]> R N , N > 2 . The non-linearity of the equation is assumed to have exponential growth in view of the logarithmic weighted Trudinger–Moser inequalities. To obtain the existence result, we apply the constrained minimization in the Nehari set, the quantitative deformation lemma and results from degree theory.

Existence of two infinite families of solutions for singular superlinear equations on exterior domains

In this article we study radial solutions of \(\Delta u + K(|x|) f(u) =0\) inthe exterior of the ball of radius \(R>0\) in \(\mathbb {R}^{N}\) with \(N>2\) where \(f\) grows superlinearly at infinity and is singular at \(0\) with \(f(u) \sim \frac{1}{|u|^{q-1}u}\) and \(0<q<1\) for small \(u\).We assume \(K(|x|) \sim |x|^{-\alpha}\) for large \(|x|\) and establish existence of two infinite families of sign-changing solutions when \(N+q(N-2) <\alpha <2(N-1)\). For more information see https://ejde.math.txstate.edu/Volumes/2024/06/abstr.html

Existence of sign-changing radial solutions with prescribed numbers of zeros for elliptic equations with the critical exponential growth in ℝ²

In this paper, we are concerned with the existence of sign-changing radial solutions with any prescribed numbers of zeros to the following Schrodinger equation with the critical exponential growth: { − Δ u + u = λ u e u 2 in R 2 , lim | x | → ∞ u ( x ) = 0 , \begin{equation*} \begin {cases} -\Delta u +u=\lambda ue^{u^2} \quad \quad \text {in } \quad \mathbb {R}^2,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0, \end{cases} \end{equation*} where 0 > λ > 1 0>\lambda >1 . Our proof relies on the shooting method, the Sturm’s comparison theorem and a Liouville type theorem in exterior domain of R 2 \mathbb {R}^2 . It seems to be the first existence result of sign-changing solution for Schrodinger equation with the critical exponential growth.

Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains

In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p,q)-Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at least γ(Ωλ∖{0}) pairs (±u) of odd weak solutions with precisely two nodal domains, where γ stands for the genus.

Sign-changing solutions for Kirchhoff-type variable-order fractional Laplacian problems

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy.

Ground-state sign-changing homoclinic solutions for a discrete nonlinear p-Laplacian equation with logarithmic nonlinearity

By using a direct non-Nehari manifold method from (Tang and Cheng in J. Differ. Equ. 261:2384–2402, 2016), we obtain an existence result of ground-state sign-changing homoclinic solutions that only changes sign once and ground-state homoclinic solutions for a class of discrete nonlinear p-Laplacian equations with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground-state energy is larger than twice the ground-state energy.