Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3

Revisit to sign-changing solutions for the nonlinear Schrödinger–Poisson system in R3

This paper is dedicated to studying the following Schrodinger–Poisson system 

$${\left\{ \begin{array}{ll}-\triangle u+V(x)u+\lambda\phi u=K(x)f(u),& \quad x\in \mathbb{R}^{3},\\-\triangle\phi= u^2,\quad x\in \mathbb{R}^{3},\end{array}\right.}$$

where V,K are positive continuous potentials, f is a continuous function and ${\lambda}$ is a positive parameter. We develop a direct approach to establish the existence of one ground state sign-changing solution ${u_\lambda}$ with precisely two nodal domains, by introducing a weaker condition that there exists ${\theta_0\in (0,1)}$ such that 

$$K(x)\left[\frac{f(\tau)}{\tau^3}-\frac{f(t\tau)}{(t\tau)^3} \right]\mathrm{sign}(1-t)+\theta_0V(x)\frac{|1-t^2|}{(t\tau)^2} \geq 0, \quad \forall x \in\mathbb{R}^3, t > 0, \tau\ne 0$$

than the usual increasing condition on ${f(t)/|t|^3}$. Under the above condition, we also prove that the energy of any sign-changing solution is strictly larger than two times the least energy, and give a convergence property of ${u_\lambda}$ as ${\lambda\searrow 0}$.

Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in $${\mathbb{R}^{3}}$$

We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schrodinger equation $$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1(\mathbb {R}^N) \end{aligned}$$
 where $$N\ge 2,$$
 $$2<p<2^*$$
 , $$\epsilon >0$$
 is a small parameter, and V is assumed to be bounded and bounded away from zero. When V has a local minimum point P, as $$\epsilon \rightarrow 0$$
 , we construct an infinite sequence of localized sign-changing solutions clustered at P and these solutions are of higher topological type in the sense that they are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. It has been an open question whether the sign-changing solutions of higher topological type can be localized and our result gives an affirmative answer. The existing results in the literature have been subject to some geometrical or topological constraints that limit the number of localized sign-changing solutions. At a local minimum point of V, Bartsch et al. (Math Ann 338:147–185, 2007) proved the existence of N pairs of localized sign-changing solutions and D’Aprile and Pistoia (Ann Inst Henri Poincare Anal Non Lineaire 26:1423–1451, 2009) constructed 9 pairs of localized sign-changing solutions for $$N\ge 3$$
 . Our result gives an unbounded sequence of such solutions. Our method combines the Byeon and Wang’s penalization approach and minimax method via a variant of the classical symmetric mountain pass theorem, and is rather robust without using any non-degeneracy conditions.

Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations

We are interested in the existence and asymptotic behavior of sign-changing solutions to the following nonlinear Schrodinger–Poisson system

 $$\left\{\begin{array}{ll}-\Delta u+V(x)u+\lambda \phi(x)u =f(u), \ &\quad x \in \mathbb{R}^3,\\ -\Delta \phi=u^2, \ &\quad x \in \mathbb{R}^3,\end{array}\right.$$
 where V(x) is a smooth function and λ is a positive parameter. Because the so-called nonlocal term $${\lambda \phi_u(x)u}$$
 is involving in the equation, the variational functional of the equation has totally different properties from the case of $${\lambda=0}$$
 . Under suitable conditions, combining constraint variational method and quantitative deformation lemma, we prove that the problem possesses one sign-changing solution $${u_\lambda}$$
 . Moreover, we show that any sign-changing solution of the problem has an energy exceeding twice the least energy, and for any sequence $${\{\lambda_n\} \rightarrow 0^+(n \rightarrow \infty)}$$
 , there is a subsequence $$\{\lambda_{n_k}\}$$
 , such that $${u_{\lambda_{n_k}}}$$
 converges in $${H^1(\mathbb{R}^3)}$$
 to $${u_0}$$
 as $${k\rightarrow \infty}$$
 , where $${u_0}$$
 is a sign-changing solution of the following equation

 $$-\Delta u+V(x)u=f(u),\quad \ x \in \mathbb{R}^3$$
 .

Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger–Poisson system in ${\mathbb{R}^3}$

In this paper, we consider the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u)&{}\quad \text{ in }\ \mathbb {R}^3,\\ -\Delta \phi =u^2&{}\quad \text{ in }\ \mathbb {R}^3. \end{array} \right. \end{aligned}$$
 We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity $$f(u)=|u|^{p-2}u$$
 for the well-studied case $$p\in (4,6)$$
 , and the less studied case $$p\in (3,4)$$
 , and for the latter case, few existence results are available in the literature.

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Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system

We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity $f(u)=|u|^{p-2}u$ for the well-studied case $p\in(4,6)$, and the less-studied case $p\in(3,4)$, and for the latter case few existence results are available in the literature.

Jianjun Zhang

Papers

Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system

Infinitely many sign-changing solutions for the nonlinear Schr\"{o}dinger-Poisson system