Existence Of Bound State Solutions Research Articles

Bound state solutions for quasilinear Schrödinger equations with Hardy potential

This paper is concerned with the existence of bound state solutions for quasilinear Schrödinger equations with Hardy potential and Berestycki–Lions type conditions. By using the variational methods, we get the existence results which is a complement to the ones in Hu et al. (2022).

Existence of a positive bound state solution for logarithmic Schrödinger equation

We investigate the existence of bound state solutions for the logarithmic Schrödinger equation−Δu+V(x)u=ulog⁡u2inRN,N≥1, where V(x)∈C(RN) has a limit at infinity. In the case when the ground state is not attained, we construct a higher critical value for the variational functional. The classical variational methods cannot be applied directly to deal with the above problem due to nonsmoothness. To recover the smoothness, we use the superlinear power-law perturbation of the logarithmic nonlinearity.

Bound States for the Spin-1/2 Aharonov-Bohm Problem in a Rotating Frame

In this paper, we study the effects of rotation in the spin-1/2 non-relativistic Aharonov-Bohm problem for bound states. We use a technique based on the self-adjoint extension method and determine an expression for the energies of the bound states. The inclusion of the spin element in the Hamiltonian guarantees the existence of bound state solutions. We perform a numerical analysis of the energies and verify that both rotation and the spin degree of freedom affect the energies of the particle. The main effect we observe in this analysis is a cutoff value manifested in the Aharonov-Bohm flux parameter that delimits the values for the positive and negative energies.

Existence of bound state solutions for the generalized Chern–Simons–Schrödinger system in [formula omitted

This paper is dedicated to studying the generalized Chern–Simons–Schrödinger system in H1(R2). By the concentration-compactness principle, a new result on the existence of bound state solutions is obtained. This conclusion extends some known results in previous papers.

Bound state solutions for some non-autonomous asymptotically cubic Schrödinger–Poisson systems

We are concerned with the following Schrodinger–Poisson systems $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+u+K(x)\phi (x)u=q(x)f(u), &{}\quad x\in {\mathbb {R}}^3, \\ -\Delta \phi =K(x)u^2, &{} \quad x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$ where f is asymptotically cubic, $$\lim _{|x|\rightarrow \infty }K(x)=0$$ and $$\lim _{|x|\rightarrow \infty }q(x)=q_{\infty }>0$$ . We establish the existence of bound state solutions to this problem by using the method developed in Szulkin and Weth [20, 21].

Mode solutions for a Klein-Gordon field in anti–de Sitter spacetime with dynamical boundary conditions of Wentzell type

We study a real, massive Klein-Gordon field in the Poincar\'e fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of dynamical boundary conditions of generalized Wentzell type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincar\'e fundamental domain of AdS. We completely solve the equations for the bulk and boundary fields and investigate the existence of bound state solutions, motivated by the analogous problem with Robin boundary conditions, which are recovered as a limiting case. Finally, we argue that both Robin and generalized Wentzell boundary conditions are distinguished in the sense that they are invariant under the action of the isometry group of the AdS conformal boundary, a condition which ensures in addition that the total flux of energy across the boundary vanishes.

Existence of a bound state solution for quasilinear Schrödinger equations

Abstract In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in ℝ N {\mathbb{R}^{N}} . After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in H 1 ⁢ ( ℝ N ) {H^{1}(\mathbb{R}^{N})} . The proofs are based on the Pohozaev manifold and a linking theorem.

Standing waves for a system of nonlinear Schrödinger equations in R N

In this paper we study the existence of bound state solutions for stationary Schrodinger systems of the form � −�u + V( x)u= K(x)Fu(u, v) in R N , −�v + V( x)v= K(x)Fv(u, v) in R N , where N 3, V and K are bounded continuous nonnegative functions, and F( u, v)is a C 1 and p-homogeneous function with 2 <p< 2N/(N − 2). We give a special attention to the case when V may eventually vanishes. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.

Fermions in the background of mixed vector–scalar–pseudoscalar square potentials

The general Dirac equation in 1+1 dimensions with a potential with a completely general Lorentz structure is studied. Considering mixed vector–scalar–pseudoscalar square potentials, the states of relativistic fermions are investigated. This relativistic problem can be mapped into a effective Schrödinger equation for a square potential with repulsive and attractive delta-functions situated at the borders. An oscillatory transmission coefficient is found and resonant state energies are obtained. In a special case, the same bound energy spectrum for spinless particles is obtained, confirming the predictions of literature. We showed that existence of bound-state solutions is conditioned by the intensity of the pseudoscalar potential, which possess a critical value.

On the existence of sign changing bound state solutions of a quasilinear equation

In this paper we establish the existence of bound state solutions having a prescribed number of sign changes for(P)Δmu+f(u)=0,x∈RN,N⩾m>1, where Δmu=∇⋅(|∇u|m−2∇u). Our result is new even for the case of the Laplacian (m=2).

Bound state solutions for a class of nonlinear Schrödinger equations

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, where V, K are positive continuous functions and p &gt; 1 is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential V is allowed to vanish at infinity and the competing function K does not have to be bounded. In the \\emph{semi-classical limit}, i.e. for \varepsilon\sim 0 , we prove the existence of bound state solutions localized around local minimum points of the auxiliary function \mathcal{A} = V^\theta K^{-\frac{2}{p-1}} , where \theta=(p+1)/(p-1)-N/2 . A special attention is devoted to the qualitative properties of these solutions as \varepsilon$ goes to zero.