Related papers: Variational methods for scaled functionals with ap…
We study critical growth elliptic problems with jumping nonlinearities. Standard linking arguments based on decompositions of $H^1_0(\Omega)$ into eigenspaces of $- \Delta$ cannot be used to obtain nontrivial solutions to such problems. We…
A variational technique is established to deal with the Schrodinger equation with parity-time(PT) symmetric Gaussian complex potential. The method is extended to the linear and self-focusing and defocusing nonlinear cases. Some unusual…
In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schr\"odinger equation. The method is based on: i) the definition of a linearly independent working…
The authors study the method of scaling in the context of the study of automorphism groups of complex domains in multiple dimensions. Various types of scaling techniques are compared and contrasted. Applications are given in a number of…
By exploiting a variational technique based upon projecting over the Pohozaev manifold, we prove existence of positive solutions for a class of nonlinear fractional Schrodinger equations having a nonhomogenous nonautonomous asymptotically…
We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$ - \Delta u + (I_\alpha \star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$ where $N \geq 2$, $I_\alpha : \mathbb{R}^N \setminus…
We generate hierarchies of derivative nonlinear Schr\"odinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established…
We develop a calculus of variations for functionals which are defined on a set of non differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the…
In this paper, we consider the following 2-D Schr\"{o}dinger-Newton equations \begin{eqnarray*} -\Delta u+a(x)u+\frac{\gamma}{2\pi}\left(\log(|\cdot|)*|u|^p\right){|u|}^{p-2}u=b{|u|}^{q-2}u \qquad \text{in} \,\,\, \mathbb{R}^{2},…
Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite…
This article motivates and presents the scale relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. It stems from the scale relativity proposal to extend the principle of relativity to…
In this paper, we combine the method of multiple scales and the method of matched asymptotic expansions to construct uniformly-valid asymptotic solutions to autonomous and non-autonomous difference equations in the neighbourhood of a…
We develop a variational method for interacting surface systems with higher-form global symmetries. As a natural extension of the conventional second-quantized Hamiltonian of interacting bosons, we explicitly construct a second-quantized…
We formulate symmetric versions of classical variational principles. Within the framework of non-smooth critical point theory, we detect Palais-Smale sequences with additional second order and symmetry information. We discuss applications…
The recent literature on first order methods for smooth optimization shows that significant improvements on the practical convergence behaviour can be achieved with variable stepsize and scaling for the gradient, making this class of…
Variational inequalities offer a versatile and straightforward approach to analyzing a broad range of equilibrium problems in both theoretical and practical fields. In this paper, we consider a composite generally non-monotone variational…
An integrable hierarchies connected with linear stationary Schr\"odinger equation with energy dependent potentials (in general case) are considered. Galilei-like and scaling invariance transformations are constructed. A symmetry method is…
Solitons of a discrete nonlinear Schr\"{o}dinger equation which includes the next-nearest-neighbor interactions are studied by means of a variational approximation and numerical computations. A large family of multi-humped solutions,…
Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schr\"odinger…
In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3,…