Radial Sign-changing Solutions Research Articles

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.

On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis–Nirenberg problem

In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis–Nirenberg problem ruled by the s-Laplacian, in a ball of Rn, when s∈(0,1) and n>6s. As usual, λ is the (positive) parameter in the linear part in u. We prove that for λ sufficiently small such solutions cannot vanish at the origin, we show that they change sign at most twice and their zeros coincide with the sign-changes. Moreover, when s is close to 1, such solutions change sign exactly once. Finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a “tower of bubbles”, as λ→0+, i.e. the positive and negative parts concentrate at the same point with different concentration speeds.

Radial ground state sign-changing solutions for a class of asymptotically cubic or super-cubic Schrödinger–Poisson type problems

This paper is dedicated to studying the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+V(|x|)u+\lambda \phi u=f(|x|,u), &{}\quad x\in \mathbb {R}^3,\\ -\triangle \phi = u^2,&{}\quad x\in \mathbb {R}^3, \end{array} \right. \end{aligned}$$ where $$\lambda $$ is a positive parameter, $$V\in {\mathcal {C}}(\mathbb {R}^{3}, (0,\infty ))$$ and $$f\in {\mathcal {C}}({\mathbb {R}}^{3}\times \mathbb {R}, \mathbb {R})$$ . Using weaker conditions $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^3}=\infty $$ uniformly in $$x\in \mathbb {R}^3$$ , and $$\begin{aligned} \left[ \frac{f(x,\tau )}{\tau ^3}-\frac{f(x,t\tau )}{(t\tau )^3}\right] {\mathrm {sign}}(1-t) +\theta _0V(x)\frac{|1-t^2|}{(t\tau )^2}\ge 0, \quad \forall \;\; x\in \mathbb {R}^3,\ t>0, \;\; \tau \ne 0 \end{aligned}$$ with a constant $$\theta _0\in (0,1)$$ , instead of the usual super-cubic condition $$\lim _{|t|\rightarrow \infty }\frac{\int _0^tf(x, s){\mathrm {d}}s}{|t|^4}=\infty $$ uniformly in $$x\in \mathbb {R}^3$$ , and the Nehari type monotonic condition on $$f(x,t)/|t|^3$$ , we establish the existence of one radial ground state sign-changing solution $$u_\lambda $$ with precisely two nodal domains. Under the same conditions, 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 $$u_\lambda $$ as $$\lambda \searrow 0$$ . Our result unifies both asymptotically cubic and super-cubic cases, which extends the existing ones in the literature.

On a Kirchhoff Equation in Bounded Domains

Abstract In this paper, we consider the following Kirchhoff equation: { - ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ u + | u | p - 2 u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}\bigg{(}a+b\int_{\Omega}\lvert\nabla u% |^{2}\,dx\bigg{)}\Delta u=\lambda u+|u|^{p-2}u&&\displaystyle\text{in }\Omega,% \\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) is a bounded domain with smooth boundary ∂ ⁡ Ω {\partial\Omega} , 2 < p < 2 * = 2 ⁢ N N - 2 {2<p<2^{*}=\frac{2N}{N-2}} is the Sobolev exponent and a, b, λ are positive parameters. By the variational method, we obtain some existence and multiplicity results of the sign-changing solutions (including the radial sign-changing solution in the case of Ω = 𝔹 R {\Omega=\mathbb{B}_{R}} ) for this problem. Some further properties of these sign-changing solutions, such as the numbers of the nodal domains, the concentration behaviors as b → 0 + {b\to 0^{+}} , the estimates of the energy values and so on, are also obtained. Our results generalize and improve some known results in the literature. Moreover, we also obtain a uniqueness result of the radial positive solution.

Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

Abstract This paper is concerned with the following type of quasilinear elliptic equations in ℝ N {\mathbb{R}^{N}} involving the p-Laplacian and critical growth: - Δ p ⁢ u + V ⁢ ( | x | ) ⁢ | u | p - 2 ⁢ u - Δ p ⁢ ( | u | 2 ) ⁢ u = λ ⁢ | u | q - 2 ⁢ u + | u | 2 ⁢ p * - 2 ⁢ u , -\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u, which arises as a model in mathematical physics, where 2 < p < N {2<p<N} , p * = N ⁢ p N - p {p^{*}=\frac{Np}{N-p}} . For any given integer k ≥ 0 {k\geq 0} , by using change of variables and minimization arguments, we obtain, under some additional assumptions on p and q, a radial sign-changing nodal solution with k + 1 {k+1} nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

Global bifurcation of sign-changing radial solutions of elliptic equations of order 2<i>m</i> in annular domains

In this paper we study the global bifurcation of sign-changing radial solutions for some semilinear elliptic problems of order 2m in an annulus with Dirichlet boundary conditions.

Morse index of radial nodal solutions of Hénon type equations in dimension two

We consider non-autonomous semilinear elliptic equations of the type [Formula: see text] where [Formula: see text] is either a ball or an annulus centered at the origin, [Formula: see text] and [Formula: see text] is [Formula: see text] on bounded sets of [Formula: see text]. We address the question of estimating the Morse index [Formula: see text] of a sign changing radial solution [Formula: see text]. We prove that [Formula: see text] for every [Formula: see text] and that [Formula: see text] if [Formula: see text] is even. If [Formula: see text] is superlinear the previous estimates become [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] denotes the number of nodal sets of [Formula: see text], i.e. of connected components of [Formula: see text]. Consequently, every least energy nodal solution [Formula: see text] is not radially symmetric and [Formula: see text] as [Formula: see text] along the sequence of even exponents [Formula: see text].

Existence results for fully nonlinear equations in radial domains

ABSTRACTWe consider the fully nonlinear problemwhere F is uniformly elliptic, p>1 and Ω is either an annulus or a ball in ℝn, n≥2.We prove the following results:existence of a positive/negative radial solution for every exponent p>1, if Ω is an annulus;existence of infinitely many sign changing radial solutions for every p > 1, characterized by the number of nodal regions, if Ω is an annulus;existence of infinitely many sign changing radial solutions characterized by the number of nodal regions if Ω is a ball and p is subcritical.

Nonexistence of least energy nodal solutions for Schrödinger–Poisson equation

In this paper, we are interested in the least energy nodal solutions for nonlinear Schrödinger–Poisson equation in R3. Because the presence of the nonlocal term λϕu(x)u makes the variational functional of this equation totally different from the case of λ=0. It is still unknown whether the least energy nodal solution in H1(R3) exists or not when λ>0, although a sign-changing radial solution of this equation has been obtained in Wang and Zhou (2015). By introducing an odd Nehari manifold, we give a negative answer of this question when p∈(3,5) and λ>0, via the so-called “energy doubling” property.

Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem

We study uniqueness of sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} \Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1 < p < \frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution. The existence of sign-changing radial solutions with exactly $k$ nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial solution is unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.

Uniqueness of sign-changing radial solutions for Δu − u + |u|p − 1u = 0 in some ball and annulus

The following Dirichlet problem{Δu−u+|u|p−1u=0inΩ,u=0on∂Ω, is considered, where Ω is either an annulus or a ball in RN and p>1. The uniqueness of radial solutions having exactly k−1 nodes is shown for the following cases: Ω is a sufficiently thin annulus; Ω is a certain small ball, N≥4 and 1<p<N/(N−2); Ω is the unit ball, N=3 and 1<p≤3; Ω is any annulus or any ball, but p>1 is sufficiently close to 1 and N=3, 5 or 7.

Exact Morse index computation for nodal radial solutions of Lane–Emden problems

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq2$, centered at the origin and $1<p<p_S$, with $p_S=+\infty$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\geq3$. Our main result is to prove that in dimension $N=2$ the Morse index of the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} is exactly $12$ if $p$ is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in $\mathbb R^N$ in any dimension $N\geq2$.

Radial sign-changing solution for fractional Schrödinger equation

We study the existence of a radial sign-changing solution for the stationary fractional Schrodinger equation

Radial sign-changing solutions to biharmonic nonlinear Schrödinger equations

In this work we obtain three radial solutions of a biharmonic stationary Schrödinger equation, one being positive, one negative, and one sign changing. The dual decomposition method is used to split the natural second-order Sobolev space considered in order to apply the appropriate variational approach.

Asymptotic analysis for radial sign-changing solutions of the Brezis–Nirenberg problem

We study the asymptotic behavior, as \(\lambda \rightarrow 0\), of least energy radial sign-changing solutions \(u_\lambda \), of the Brezis–Nirenberg problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = \lambda u + |u|^{2^* -2}u &{}\quad \hbox {in} B_1 u=0 &{}\quad \hbox {on} \partial B_1, \end{array}\right. \end{aligned}$$ where \(\lambda >0,\, 2^*=\frac{2n}{n-2}\) and \(B_1\) is the unit ball of \(\mathbb {R}^n,\, n\ge 7\). We prove that both the positive and negative part \(u_\lambda ^+\) and \(u_\lambda ^-\) concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover, we show that suitable rescalings of \(u_\lambda ^+\) and \(u_\lambda ^-\) converge to the unique positive regular solution of the critical exponent problem in \(\mathbb {R}^n\). Precise estimates of the blow-up rate of \(\Vert u_\lambda ^\pm \Vert _{\infty }\) are given, as well as asymptotic relations between \(\Vert u_\lambda ^\pm \Vert _{\infty }\) and the nodal radius \(r_\lambda \). Finally, we prove that, up to constant, \(\lambda ^{-\frac{n-2}{2n-8}} u_\lambda \) converges in \(C_{\mathrm{loc}}^1(B_1-\{0\})\) to \(G(x,0)\), where \(G(x,y)\) is the Green function of the Laplacian in the unit ball.

Critical exponents and critical dimensions for nonlinear elliptic problems with singular coefficients

Let B1 ⊂ ℝN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients: { u|∂ B1 = 0,−div(|∇u|p−2∇u)=|x|s|u|p*(s)−2u+λ|x|t|u|p−2u, x ∈ B1,where t,s > −p, 2 ≤p < N, p*(s)=(N+s)pN−p and λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N > p(p−1)t + p(p2−p+1) and λ ∈ (0, λ1, t), where λ1, t is the first eigenvalue of -Δp with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ (ps+p)min{1,p+tp+s}+p2p−(p−1)min{1,p+tp+s} λ > 0 is small.

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

Asymptotic behaviour of sign changing radial solutions of Lane Emden Problems in the annulus

We study the asymptotic behaviour as $p\rightarrow \infty$ of the nodal radialsolutions $u_p$ of the problem\begin{equation*}\left\{\begin{array}{rlll}-\Delta u&=&|u|^{p-1}u& \text{in }\Omega \\u&=&0& \text{on }\partial\Omega,\end{array}\right.\end{equation*}where $\Omega$ is an annulus in $\mathbb{R}^N$, $N\geq 2$.We also analyze the spectrum of the linearized operator associated to $u_p$in the case when $u_p$ has only two nodal regions. In particular, we prove that the Morse index of $u_p$ tends to $\infty$ as $p$ goes to $\infty$.

Non-existence of radial backward self-similar blow-up solutions with sign change

We consider a Cauchy problem for a semilinear heat equationwith p &gt; 1. If u(x, t) = (T − t)−1/(p−1)ϕ((T − t)−1/2x) for x ∈ ℝN and t ∈ [0, T),where ϕ ∈ L∞(ℝN) is a solution not identically equal to zero ofthen u is called a backward self-similar solution blowing up at t = T. We show that, for all p &gt; 1, there exists no radial sign-changing solution of (E) which belongs to L∞(ℝN). This implies the non-existence of radial backward self-similar solution with sign change blowing up in finite time.

Poincaré–Sobolev equations in the hyperbolic space

We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais–Smale characterisation of the problem $${-\Delta_{{\mathbb B}^{N}}u - \lambda u = |u|^{p-1}u, u\in H^1({\mathbb B}^{N})}$$ in the hyperbolic space $${{\mathbb B}^{N}}$$ where $${1 < p\leq\frac{N+2}{N-2}}$$ . We will also prove the existence of sign changing solution to the Hardy–Sobolev–Mazya equation and the critical Grushin problem.