Radial Sign-changing Solutions Research Articles

A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions

In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system{−Δuj+λjuj=μjuj3+∑i≠jβijui2ujin B1,uj∈H0,r1(B1) for j=1,⋯,N. Here, λj,μj>0 and βij=βji are constants for i,j=1,⋯,N and i≠j. B1 denotes the unit ball in the Euclidean space R3 centred at the origin. For any P1,⋯,PN∈N, we prove the uniqueness of the radial solution (u1,⋯,uj) with uj changes its sign exactly Pj times for any j=1,⋯,N in the following case: λj≥1 and |βij| are small for i,j=1,⋯,N and i≠j. New Liouville-type theorems and boundedness results are established for this purpose.

Azer Sign-changing radial solutions for a semilinear problem on exterior domains with nonlinear boundary conditions

In this paper we are interested to the existence and multiplicity of sign changing radial solutions of problem of elliptic equations $\Delta U(x)+\varphi(|x|)f(U)=0$ with a nonlinear boundary conditions on exterior of the unite ball centered at the origin in $\mathbb{R}^{N}$ such that $ u(x) \rightarrow 0$ as $ |x|\to \infty $, with any given number of zeros where the nonlinearity $ f(u) $ is odd, superlinear for $ u $ lager enough and $ f0 $ on $(0,\beta)$, $ f0$ on $(\beta,\infty) $. The function $\varphi0$ is $ C^{1} $ on $ [R,\infty) $ where $ 0\varphi(|x|)\leq c_0\,|x|^{-\alpha}$ with $ \alpha2(N-1) $ and $ N2 $ for large $ |x| $.

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.

Existence of radial sign-changing solutions for fractional Kirchhoff-type problems in R3

Existence of radial sign-changing solutions for fractional Kirchhoff-type problems in R3

Sign-changing solutions for Schrödinger–Poisson system with p-Laplacian in [formula omitted

In this paper, we consider the following quasilinear Schrödinger–Poisson system −Δpu+|u|p−2u+ϕu=f(u),x∈R3,−Δϕ=u2,x∈R3,where 43<p≤125, Δpu=div(|∇u|p−2∇u) and f∈C(R,R). Based on the method of invariant sets of descending flow, this system admits at least one radial sign-changing solution. If f is odd, this system admits infinitely many radial sign-changing solutions. We improve and generalize the results obtained by Liu et al. (2016) to a more general system with p-Laplacian.

Existence of sign-changing solutions for a gauged nonlinear Schrödinger equation with a quintic term

In this paper, we study the existence and asymptotic behavior of sign-changing solutions with arbitrary number of nodes to the following gauged nonlinear Schrödinger equation(0.1){−Δu+ωu+((hu(|x|))2|x|2+∫|x|+∞hu(s)su2(s)ds)u=λ|u|4uin R2,u∈H1(R2), where ω,λ>0 and hu(s)=12∫0sru2(r)dr is the so-called Chern-Simons term. The nonlocal term g(u):=((hu(|x|))2|x|2+∫|x|+∞hu(s)su2(s)ds)u is quintic in the sense that g(tu)=t5g(u), which is in complicated competition with the nonlinear term λ|u|4u. This causes that not all functions in H1(R2) can be projected onto the usual Nehari manifold, and so the classical Nehari manifold method is not applicable. By taking advantage of the Miranda theorem and Gersgorin disc theorem, via the variational method and gluing approach, we prove that for each positive integer k, the problem (0.1) admits a radial sign-changing solution Uk,6λ that changes sign exactly k times. Moreover, its energy is strictly increasing in k, and for any sequence {λn} with λn→+∞, up to a subsequence, λn14Uk,6λn converges to some U¯k,60∈H1(R2) which is a radial sign-changing solution with exactly k nodes to the following local equation{−Δu+ωu=|u|4uin R2,u∈H1(R2). Our results extend the existing ones of [6] from the super-quintic case to the quintic case.

Nodal solutions for an asymptotically linear Kirchhoff-type problem in ℝ N

ABSTRACT In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems where , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for and b large enough.

Infinitely many radial and non-radial sign-changing solutions for Schrödinger equations

Abstract In the present paper, a class of Schrödinger equations is investigated, which can be stated as − Δ u + V ( x ) u = f ( u ) , x ∈ ℝ N . - \Delta u + V(x)u = f(u),\;\;\;\;x \in {{\rm{\mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.

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.

Sign-changing solutions for critical equations with Hardy potential

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 3$, with $0 \in \Omega$: $$ \left\{ \begin{array}{ll}-\Delta u-\gamma \frac{u}{|x|^2}-\epsilon u=|u|^{\frac{4}{N-2}}u &\hbox{in }\Omega u=0 & \hbox{on }\partial \Omega, \end{array}\right. $$ when $\epsilon>0$ is small and $\gamma< {(N-2)^2\over4}$. Setting $ \gamma_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $\gamma\leq \frac{(N-2)^2}{4}-1$ and $\gamma \neq \gamma_j$ for any $j$, then for small $\epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $\gamma<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the equation has for small enough $\epsilon$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $\gamma\neq \gamma_j$ is not necessary. Indeed, it is known that, if $\gamma > \frac{(N-2)^2}{4}-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $\epsilon>0$ small. We complete the picture here by showing that, if $\gamma\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $\epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $\gamma=0$.

Multiple Sign-Changing Solutions for a Class of Schrödinger Equations with Saturable Nonlinearity

In this paper, we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics. More precisely, for any given nonnegative integer k, by using a minimization argument, we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem, and show, by a deformation lemma and Miranda’s theorem, that the minimizer is the desired solution.

Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight

<p style='text-indent:20px;'>We prove the existence of infinitely many sign changing radial solutions for a <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary.

Sign-Changing Solutions for Chern–Simons–Schrödinger Equations with Asymptotically 5-Linear Nonlinearity

In this paper, we study the following Chern–Simons–Schrodinger equation $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$ where $$\omega ,\lambda >0$$ and $$h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r$$ . Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as $$\lambda \rightarrow 0$$ .

Sign-changing solutions for nonlinear Schrödinger–Poisson systems with subquadratic or quadratic growth at infinity

This paper is dedicated to investigating the existence of sign-changing solutions to the following Schrödinger–Poisson system −Δu+V(x)u+λϕu=f(u)inR3,−Δϕ=u2inR3,where λ>0 and f is subquadratic or quadratic at infinity. By using the method of invariant sets of descending flow, we prove that the problem above admits multiple radial sign-changing solutions in the subquadratic case as λ small. As for the quadratic case, by virtue of a perturbation argument, we show that this problem admits at least one sign-changing solution as λ small.

New concentration phenomena for a class of radial fully nonlinear equations

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.

Nodal solutions for the Schrödinger–Poisson equations with convolution terms

This paper deals with the following nonlinear Schrödinger–Poisson system with convolution terms: (SPC)−Δu+V(|x|)u+bϕu=Iα∗|u|p|u|p−2uinR3,−Δϕ=u2inR3,where b>0 is a parameter, V∈C([0,∞),R+),α∈(0,3), Iα:R3→R is the Riesz potential and p∈(3+α3,3+α). The presence of nonlocal terms ϕu and Iα∗|u|p|u|p−2u makes the variational functional of (SPC) totally different from the case of b=0 or the case with pure power nonlinearity. Taking advantage of the results from the matrix theory and the Brouwer degree theory, we introduce some new analytic techniques to prove that for any given integer k≥1, (SPC) admits a sign changing radial solution ukb for p>4, which changes sign exactly k times. Furthermore, for any sequence {bn} with bn→0+ as n→∞, there is a subsequence, still denoted by {bn}, such that ukbn converges to uk0 in H1(R3) as n→∞, where uk0 also changes sign exactly k times and is a sign-changing radial solution of the Choquard equation −Δu+V(|x|)u=Iα∗|u|p|u|p−2uinR3.Our result generalizes the existing ones for the Schrödinger–Poisson equations and Choquard equations, and seems to be the first result of such radial solutions for an equation with two competing convolution terms. Besides, we show that the degeneracy for this existence result happens for p<2.

Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth

We investigate the following gauged nonlinear Schrodinger equation \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+\omega u+\lambda\bigg(\dfrac{h_{u}^{2}(|x|)}{|x|^{2}}+ \int_{|x|}^{+\infty}\dfrac{h_{u}(s)}{s}u^{2}(s)ds\bigg)u = f(u) \mbox{in} \mathbb{R}^{2}, u\in H_r^1(\mathbb{R}^{2}), \end{cases} \end{equation*} $\end{document} where \begin{document}$ \omega,\lambda>0 $\end{document} and \begin{document}$ h_{u}(s) = \frac{1}{2}\int_{0}^{s}ru^{2}(r)dr $\end{document} . When \begin{document}$ f $\end{document} has exponential critical growth, by using the constrained minimization method and Trudinger-Moser inequality, it is proved that the equation has a ground state radial sign-changing solution \begin{document}$ u_{\lambda} $\end{document} which changes sign exactly once. Moreover, the asymptotic behavior of \begin{document}$ u_{\lambda} $\end{document} as \begin{document}$ \lambda\rightarrow0 $\end{document} is analyzed.

Infinitely many radial solutions for a sub-super critical p-Laplacian problem

We prove the existence of infinitely many sign-changing radial solutions for a p-sub-super critical p-Laplacian Dirichlet problem in a ball. We consider an equation defined by the p-Laplacian operator perturbed by a nonlinearity g(u) that is p-subcritical at $$+\,\infty $$ and p-supercritical at $$-\,\infty $$. Our results extend those in Castro et al. (Electron. J. Differ. Equ. 2007(111):1–10, 2007) for the corresponding semilinear case and those of El Hachimi and De Thelin (J. Differ. Equ. 128:78–102, 1996) where the subcritical case was studied.

Radial ground state sign-changing solutions for asymptotically cubic or super-cubic fractional Schrödinger-Poisson systems

ABSTRACTIn this paper, we consider the following fractional Schrödinger-Poisson system: where , λ is a positive parameter and f satisfies asymptotically cubic or super-cubic growth that is very different from super-cubic growth in previous literatures. Without assuming the usual Nehari-type monotonic condition on , we establish the existence of one radial ground state sign-changing solution with precisely two nodal domains. Furthermore, we also prove that the energy of any radial sign changing solution is strictly larger than two times the least energy and give a convergence property of as . Our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for s=t=1.

Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions

This paper is dedicated to studying the following fractional Kirchhoff-type equation $$\begin{aligned} \left( a+b\int _{\mathbb {R}^N}|(-\triangle )^{\alpha /2}{u}|^2\mathrm {d}x\right) (-\triangle )^{\alpha }u+V(|x|)u =f(|x|, u), \ \ \ \ x\in {\mathbb {R}}^{N}, \end{aligned}$$where \(a, b>0\), either \(N=2\) and \(\alpha \in (1/2,1)\) or \(N=3\) and \(\alpha \in (3/4,1)\) holds, \(V\in \mathcal {C}(\mathbb {R}^{N}, [0,\infty ))\) and \(f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})\). By combining the constraint variational method with some new inequalities, we prove that the above problem possesses a radial sign-changing solution \(u_b\) for \(b\ge 0\) without the usual Nehari-type monotonicity condition on f, and its energy is strictly larger than twice that of the ground state radial solutions of Nehari-type. Moreover, we establish the convergence property of \(u_b\) as \(b\searrow 0\). In particular, our results unify both asymptotically cubic and super-cubic cases, which improve and complement the existing ones in the literature.