Sign-changing Solutions Research Articles

Remarks on a class of integro-differential problems

A class of variational elliptic problems involving integro-differential diffusion and Dirichlet boundary condition is studied. The absence of a maximum principle is observed and sign-changing solutions are obtained when the nonlinearity is a sublinear or superlinear power of u+, depending on the size of a positive parameter multiplying the nonlocal term. More general integro-differential operators, for which the problem is no longer variational, are also considered.

Ground State Sign-Changing Solutions for a Kirchhoff Equation with Asymptotically 3-Linear Nonlinearity

Ground State Sign-Changing Solutions for a Kirchhoff Equation with Asymptotically 3-Linear Nonlinearity

Least energy sign-changing solutions of fractional Kirchhoff–Schrödinger–Poisson system with critical and logarithmic nonlinearity

In the present paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system with logarithmic and critical nonlinearity: where and Ω is a bounded domain in with Lipschitz boundary. Combining constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has a least energy sign-changing solution . Moreover, we show that the energy of is strictly larger than two times the ground state energy. Finally, we regard b as a parameter and show the convergence property of as .

A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity

By employing a nonlocal perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form $$\begin{aligned} -\left( a+ b\int _{\mathbb {R}^3}|\nabla u|^2\right) \Delta {u}+V(x)u=f(u),\,\,x\in \mathbb {R}^3, \end{aligned}$$ where $$a,b>0$$ are constants, $$V\in C(\mathbb {R}^3,\mathbb {R})$$ , $$f\in C(\mathbb {R},\mathbb {R})$$ . The methodology proposed in the current paper is robust, in the sense that, neither the monotonicity condition on f nor the coercivity condition on V is required. Our result improves the study made by Deng et al. (J Funct Anal 269:3500–3527, 2015), in the sense that, in the present paper, the nonlinearities include the power-type case $$f(u)=|u|^{p-2}u$$ for $$p\in (2,4)$$ , in which case, it remains open in the existing literature whether there exist infinitely many sign-changing solutions to the problem above. Moreover, energy doubling is established, namely, the energy of sign-changing solutions is strictly larger than two times that of the ground state solutions for small $$b>0$$ .

Infinitely many sign-changing solutions to a conformally invariant integral equation on [formula omitted

In this work we consider the Euler–Lagrange equation for the critical points of a functional related to the conformal exponent case of the classical Hardy–Littlewood–Sobolev inequality. By lifting the problem to the sphere and leveraging the symmetries of the lifted problem, the existence of an unbounded sequence of sign-changing solutions is established. As a consequence, the existence of an unbounded sequence of sign-changing solutions to a non linear system of equations involving the fractional Laplacian is obtained.

Sign-changing solutions for a kind of Klein–Gordon–Maxwell system

In this paper, the subcritical Klein–Gordon–Maxwell system −Δu + V(x)u − (2ω + ϕ)ϕu = f(u) coupled with Δϕ=(ω+ϕ)u2,x∈R3 is studied. Under general assumptions on V, f, by using the abstract critical theorems, which are derived from the minimax method and the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions for this system, in particular, a sequence of high energy sign-changing solutions is obtained if f is odd.

Existence of least energy positive and nodal solutions for a quasilinear Schrödinger problem with potentials vanishing at infinity

In this paper, we prove the existence of at least two nontrivial solutions for a class of quasilinear problems with two non-negative and continuous potentials. Thanks to the geometries of these potentials, we are able to prove compact embeddings in some weighted Sobolev spaces, and by a minimization argument, we find a positive and a nodal (or sign-changing) (weak) solution with two nodal domains or that changes the sign exactly once in RN for such problems. The nonlinearity in this problem satisfies suitable growth and monotonicity conditions, which allow this result to complement the classical results due to Liu, Wang, and Wang [Commun. Partial Differ. Equations 29, 879–901 (2004)].

Least energy sign-changing solutions for a class of fractional Kirchhoff–Poisson system

Using the constraint variational method and a quantitative deformation lemma, we establish the existence of the least energy sign-changing solutions for the fractional Kirchhoff–Poisson system, a+b∫R3|(−Δ)s2u|2dx(−Δ)su+V(x)u+ϕ(x)u=f(x,u),(−Δ)tϕ=u2,x∈R3, where a > 0 is a constant, b∈R+ is a parameter, s, t ∈ (0, 1) and 4s + 2t > 3, (−Δ)s stands for the fractional Laplacian, V is a continuous, positive function, and f is nonlinear function satisfying suitable growth assumptions. Moreover, for any b > 0, we prove that the energy of the least energy sign-changing solution is strictly larger than twice the ground state energy. Furthermore, we show a convergence property of the least energy sign-changing solutions as the parameter b goes to zero. Our results complement an in-depth study of Wang, Radulescu, and Zhang [J. Math. Phys. 60, 011506 (2019)] in the sense that we are concerned with the nodal characteristics of the ground states.

Symmetry Properties of Sign-Changing Solutions to Nonlinear Parabolic Equations in Unbounded Domains

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.

Sufficient and necessary conditions for ground state sign-changing solutions to the Schrödinger–Poisson system with cubic nonlinearity on bounded domains

In this paper we are interested in the existence of ground state sign-changing solutions to the following Schrödinger–Poisson system −Δu+ϕu=λu+μ|u|2uinΩ−Δϕ=u2inΩu=ϕ=0on∂Ω,where Ω is a bounded smooth domain of R3, μ>0, λ<λ1 and λ1 is the first eigenvalue of −Δ,H01(Ω). Using construction technique, we show that the above system possesses one ground state sign-changing solution if and only if μ>0, which improves the recent results by Khoutir (2021).

Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

We investigate the following problem -div(v(x)|∇u|m-2∇u)+V(x)|u|m-2u=|x|-θ∗|u|b|x|α|u|b-2|x|αu+λ|x|-γ∗|u|c|x|β|u|c-2|x|βuinRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\mathrm{div}(v(x)|\ abla u|^{m-2}\ abla u)+V(x)|u|^{m-2}u= \\left( |x|^{-\ heta }*\\frac{|u|^{b}}{|x|^{\\alpha }}\\right) \\frac{|u|^{b-2}}{|x|^{\\alpha }}u+\\lambda \\left( |x|^{-\\gamma }*\\frac{|u|^{c}}{|x|^{\\beta }}\\right) \\frac{|u|^{c-2}}{|x|^{\\beta }}u \\quad \ ext { in }{\\mathbb {R}}^{N}, \\end{aligned}$$\\end{document}where b,c,α,β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b, c, \\alpha , \\beta >0$$\\end{document}, θ,γ∈(0,N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta ,\\gamma \\in (0,N)$$\\end{document}, N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 3$$\\end{document}, 2≤m<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le m< \\infty$$\\end{document} and λ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\in {\\mathbb {R}}$$\\end{document}. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

Infinitely many nodal solutions with a prescribed number of nodes for the Kirchhoff type equations

In this paper, we investigate the existence of multiple radial sign-changing solutions with the nodal characterization for a class of Kirchhoff type problems{−(a+b|∇u|L22)Δu+V(|x|)u=K(|x|)f(u)in RN,u∈H1(RN), where N=1,2,3,a,b>0, V,K are radial and bounded away from below by positive numbers. Under some weak assumptions on f∈C0(R;R), by taking advantage of the Gersgorin disc's theorem and Miranda theorem, we develop some new analytic techniques and prove that this problem admits infinitely many nodal solutions {Ukb} having a prescribed number of nodes k, whose energy is strictly increasing in k. Moreover, the asymptotic behaviors of Ukb as b→0+ are established. These results improve and generalize the previous results in the literature.

Infinitely many sign-changing solutions of a critical fractional equation

In this paper, we obtain results of nonexistence of nonconstant positive solutions, and also existence of an unbounded sequence of sign-changing solutions for some critical problems involving conformally invariant operators on the unit sphere, in particular to the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together with blow up arguments, a variant of Pohozaev’s type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere.

Multiplicity and asymptotic behavior of solutions for quasilinear elliptic equations with small perturbations

This paper considers the following general form of quasilinear elliptic equation with a small perturbation:{−∑i,j=1NDj(aij(x,u)Diu)+12∑i,j=1NDtaij(x,u)DiuDju=f(x,u)+εg(x,u),x∈Ω,u∈H01(Ω), where Ω⊂RN(N≥3) is a bounded domain with smooth boundary and |ε| small enough. We assume the main term in the equation to have a mountain pass structure but do not suppose any conditions for the perturbation term εg(x,u). Then we prove the equation possesses a positive solution, a negative solution and a sign-changing solution. Moreover, we are able to obtain the asymptotic behavior of these solutions as ε→0.

On sign-changing solutions for quasilinear Schrödinger-Poisson system with critical growth

In this paper, we consider the following quasilinear Schrödinger-Poisson system with critical growth: where , is a smooth potential function and g is a appropriate nonlinear function. For the sake of overcoming the technical difficulties caused by the quasilinear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. Moreover, when g satisfying suitable assumptions and sufficiently large μ, we take advantage of constraint variational method, the quantitative deformation lemma, Moser iteration and approximation technique to obtain a least-energy sign-changing solution , which has precisely two nodal domains.

On the Hölder regularity of signed solutions to a doubly nonlinear equation

We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is∂t(|u|p−2u)−Δpu=0,p>1. The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.

Sign-changing solutions for modified Schrödinger–Poisson system with general nonlinearity

This paper is concerned with the existence and multiplicity of sign-changing solutions for the following modified Schrödinger–Poisson system: where is coercive, the nonlinear term f is 4-superlinear at infinity but does not need any increasing condition. By using the method of invariant sets of descending flow, the existence and multiplicity of sign-changing solutions are obtained. Furthermore, we consider the asymptotical behavior of solutions with respect to the parameter κ. These results extend and complement some existing ones in the literature.

Least energy sign-changing solution for N-Laplacian problem with logarithmic and exponential nonlinearities

In this paper, we establish sufficient conditions for the existence result of least energy sign-changing weak solution to the following quasilinear nonhomogeneous N-Laplacian equation with logarithmic and exponential nonlinearities:{−ΔNu=|u|p−2uln⁡|u|2+μf(u),inΩ,u=0,on∂Ω, where f(t) behaves like eα|t|N/(N−1). By using a main tool of constrained minimization in Nehari manifold, we consider both subcritical and critical exponential growths.

Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}} u+\lambda \, \Delta u \ge |u|^{p(x)}, \quad x\in {\mathbb {R}}^N, \end{aligned}$$where \(N\ge 1\), \(\alpha \in (0,2)\), \(\lambda \in {\mathbb {R}}\) is a constant, \(p: {\mathbb {R}}^N\rightarrow (1,\infty )\) is a measurable function, and \((-\Delta )^{\frac{\alpha }{2}}\) is the fractional Laplacian operator of order \(\frac{\alpha }{2}\). A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with variable-exponent nonlinearities. Besides the consideration of variable-exponent nonlinearities, the novelty of this work consists in investigating sign-changing solutions to the considered problems. Namely, to the best of our knowledge, only nonexistence results of positive solutions to fractional elliptic problems were investigated previously. Our approach is based on the nonlinear capacity method combined with a pointwise estimate of the fractional Laplacian of some test functions, which was derived by Fujiwara (Math Methods Appl Sci 41:4955–4966, 2018) [see also Dao and Reissig (A blow-up result for semi-linear structurally damped \(\sigma \)-evolution equations, arXiv:1909.01181v1, 2019)]. Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.

Sign-changing solutions for the boundary value problem involving the fractional $p$-Laplacian

In the paper, we consider the following boundary value problem involving the fractional $p$-Laplacian: \begin{equation} \tag{$\mathcal{P}$} \begin{cases} (-\triangle)_p^su(x)=f(x,u) &\text{in } \Omega, u(x)=0 &\text{in } \mathbb{R}^N\setminus\Omega. \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 1$, $(-\Delta)_p^{s}$ is the fractional $p$-Laplacian with $s\in (0,1)$, $p\in(1,{N}/{s})$, $f(x, u)\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$. Under the improved subcritical polynomial growth condition and other conditions, the existences of a least-energy sign-changing solution for the problem $(\mathcal{P})$ has been established.