Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent

Abstract

Abstract. We investigate existence and qualitative behaviour of solutions to nonlinear Schrödinger equations with critical exponent and singular electromagnetic potentials. We are concerned with with magnetic vector potentials which are homogeneous of degree −1, including the Aharonov-Bohm class. In particular, by variational arguments we prove a result of multiplicity of solutions distinguished by symmetry properties.

On Caffarelli-Kohn-Nirenberg Inequalities for Block-Radial Functions

“…k ·) instead of u k , which allows us, without loss of generality, to assume that j (1) k = 0 and |y (1) k | ≥ ǫ. Assume first that y (1) k has a bounded subsequence, and, therefore, a renamed subsequence that converges to some point y = 0.…”

“…k ·) instead of u k , which allows us, without loss of generality, to assume that j (1) k = 0 and |y (1) k | ≥ ǫ. Assume first that y (1) k has a bounded subsequence, and, therefore, a renamed subsequence that converges to some point y = 0. Then u k (·+y) ⇀ w (1) , and, by the symmetry of u k , this means that u k (· + ωy) ⇀ w (1) for any ω ∈ SO(γ 1 ) × · · · × SO(γ m ).…”

“…1 Introduction. 1.1 Subject of the paper Let N > 1. A function f on R N is called block-radial, or multi-radial, or m-radial, m > 1, if R N is considered as a product R γ 1 × · · · × R γm , m ∈ N, and f = f (r 1 , .…”

Ground State Solutions for Schrödinger Problems with Magnetic Fields and Hardy-Sobolev Critical Exponents

“…Motivated by the seminal work [1], some papers dealt with the Schrödinger equations with nonsingular magnetic field, for example, [2][3][4][5][6][7][8][9][10][11] and references therein. For the singular magnetic potential, we refer to [12,13]. There are a few works about the nonsingular magnetic problems with critical exponents, such as [9,14,15] for Sobolev critical exponent and [16] for Hardy critical exponent.…”

Resultados de multiplicidade para equações de Schrödinger com campo magnético via teoria de Morse e topologia do domínio

“…Resultados de existência para o caso magnético podem ser encontrados em [1], [3], [7], [15], [16], [17], [18], [19], [20], [21], [26], [28], [29], [30], [34], [37], [38], [39]. Em [3], os autores provaram que se f é uma função superlinear com crescimento subcrítico, então para λ > 0 suficientemente grande, o número de soluções não triviais do problema de Dirichlet para a equação (3) é pelo menos a categoria de Ω.…”