On a Kirchhoff type problems with potential well and indefinite potential

In this paper, we study the following Kirchhoff type problem:% \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\alpha,\lambda})}% where $4<p<6$, $\alpha$ and $\lambda$ are two positive parameters, a_0\in\bbr is a (possibly negative) constant and $a(x)\geq0$ is the potential well. By the variational method, we investigate the existence of nontrivial solutions to $(\mathcal{P}_{\alpha,\lambda})$. To our best knowledge, it is the first time that the nontrivial solution of the Kirchhoff type problem is found in the indefinite case. We also obtain the concentration behaviors of the solutions as $\lambda\to+\infty$.Comment: 1

Ground state solution of a nonlocal boundary-value problem

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a general $4-$superlinear condition on the nonlinearity $f$, we prove the existence of a ground state solution; that is a nontrivial solution which has least energy among the set of nontrivial solutions. In case which $f$ is odd with respect to the second variable, we also obtain the existence of infinitely many solutions. Under our assumptions the Nehari manifold does not need to be of class $\mathcal{C}^1$.Comment: 8 page

Multiple solutions for a class of nonhomogeneous fractional Schr\"odinger equations in RN\mathbb{R}^{N}

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $k$ is a bounded positive function, $h\in L^{2}(\mathbb{R}^{N})$, $h\not \equiv 0$ is nonnegative and $f$ is either asymptotically linear or superlinear at infinity.\\ By using the $s$-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that $|h|_{2}$ is sufficiently small

Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations on R

The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameter λ on RN is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and λ>0 is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions

Stationary States of NLS on Star Graphs

We consider a generalized nonlinear Schr\"odinger equation (NLS) with a power nonlinearity |\psi|^2\mu\psi, of focusing type, describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the \delta potential of strength \alpha on the line, including as a special case (\alpha=0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (\alpha0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity \mu<2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph.Comment: Revised version, 5 pages, 2 figure

Variational properties and orbital stability of standing waves for NLS equation on a star graph

We study standing waves for a nonlinear Schr\"odinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $i \frac{d}{dt}\Psi_t = H \Psi_t - | \Psi_t |^{2\mu} \Psi_t$, where $H$ is the Hamiltonian operator which generates the linear Schr\"odinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $\omega > \frac{\alpha^2}{N^2}$. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed $\omega$ if the nonlinearity is subcritical or critical, and for $\omega<\omega^\ast$ otherwise.Comment: 36 pages, 2 figures, final version appeared in JD