Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

<p style='text-indent:20px;'>This paper is concerned with the following quasilinear Schrödinger system in the entire space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{N}(N\geq3) $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &amp;-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ &amp; u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \alpha,\beta&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 2&lt;\alpha+\beta&lt;2^* = \frac{2N}{N-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ k &gt;0 $\end{document}</tex-math></inline-formula> is a parameter. By using the principle of symmetric criticality and the moser iteration, for any given integer <inline-formula><tex-math id="M5">\begin{document}$ \xi\geq2 $\end{document}</tex-math></inline-formula>, we construct a non-radially symmetrical nodal solution with its <inline-formula><tex-math id="M6">\begin{document}$ 2\xi $\end{document}</tex-math></inline-formula> nodal domains. Our results can be looked on as a generalization to results by Alves, Wang and Shen (Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259 (2015) 318-343).</p>

Symmetric And Nonsymmetric Solutions For a Class of Quasilinear Schrödinger Equations

We use the mountain-pass theorem combined with the principle of symmetric criticality to establish multiplicity of solutions for the class of quasilinear elliptic equations-Δu + V(z)u - Δ(uwhere N ≥ 4, the potential V : ℝ

MULTIPLE SOLUTIONS FOR A QUASILINEAR ELLIPTIC VARIATIONAL SYSTEM ON STRIP-LIKE DOMAINS

We consider the quasilinear elliptic variational system\begin{alignat*} {2} -\Delta_pu\amp=\lambda F_u(x,u,v)+\mu H_u(x,u,v) \quad\amp \amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)+\mu H_v(x,u,v) \amp \amp \text{in }\varOmega, \\ u\amp=v=0 \amp \amp \text{on }\partial\varOmega, \end{alignat*}where $\varOmega$ is a strip-like domain and $\lambda$ and $\mu$ are positive parameters. Using a recent two-local-minima theorem and the principle of symmetric criticality, existence and multiplicity are proved under suitable conditions on $F$.

EXISTENCE OF TWO NON-TRIVIAL SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC VARIATIONAL SYSTEMS ON STRIP-LIKE DOMAINS

In this paper we study the multiplicity of solutions of the quasilinear elliptic system\begin{equation} \left. \begin{aligned} -\Delta_pu\amp=\lambda F_u(x,u,v)\amp\amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)\amp\amp\text{in }\varOmega, \\ u=v\amp=0\amp\amp\text{on }\partial\varOmega, \end{aligned} \right\} \end{equation} \tag{S$_\lambda$}where $\varOmega$ is a strip-like domain and $\lambda>0$ is a parameter. Under some growth conditions on $F$, we guarantee the existence of an open interval $\varLambda\subset(0,\infty)$ such that for every $\lambda\in\varLambda$, the system (S$_\lambda$) has at least two distinct, non-trivial solutions. The proof is based on an abstract critical-point result of Ricceri and on the principle of symmetric criticality.

Free elasticae and Willmore tori in warped product spaces

We use the principle of symmetric criticality to connect the Willmore variational problem for surfaces in a warped product space with base a circle, and the free elastica variational problem for curves on its fiber. In addition we obtain a rational oneparameter family of closed helices in the anti De Sitter 3-space which are critical points of the total squared curvature functional. This means they are free elasticae. Also they are spacelike; this allows us to construct a corresponding family of spacelike Willmore tori in a certain kind of spacetime close to the Robertson-Walker spaces.