Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Abstract

Abstract. In this work we consider the magnetic NLS equationwhere N ≥ 3, A : R N → R N is a magnetic potential, possibly unbounded, V : R N → R is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : R N → C to (0.1), under conditions on the nonlinearity which are nearly optimal.

Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments

“…Here also, it can be shown (see [10]) that the least energy levels coincide for complex and real-valued solutions of (1). Since the pioneering works [2,8], it is known that the stability/instability of the standing waves is closely linked to additional variational characterizations that the associated ground states enjoy.…”

“…Moreover it is easily deduced from the proofs in [3,4] that this ϕ is still a least energy solution of (1) when the infimum is, as in (4), taken over the set of all complex-valued solutions. See [10] for a proof of this statement along with a description of the ground states as being of the form U = e iθŨ , where θ ∈ R andŨ is a real positive ground state solution of (1).…”

“…After that, the difficulty is to get rid of the Lagrange multiplier. For each choice of (α, β), long computations are involved to prove that the Lagrange multiplier is 0 and to conclude that the obtained function is in fact a solution of (10).…”

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

“…Existence and multiplicity of semiclassical solutions are given e.g. in [3,[6][7][8][9]16]. Very recently, Abatangelo and Terracini obtained existence results for problems with critical nonlinearity and singular magnetic and electric potentials [1].…”

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.…”