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### Open Access Journals

DCDS

The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case

$\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$ |

where

,

,

,

,

,

is a magnetic potential,

is an electric potential,

is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.

$s∈ (0,1)$ |

$N>2s$ |

$a∈ \mathbb{R}^+_0$ |

$b∈ \mathbb{R}^+_0$ |

$θ∈[1,N/(N-2s))$ |

$A:\mathbb{R}^N\to\mathbb{R}^N$ |

$V:\mathbb{R}^N\to \mathbb{R}^+$ |

$(-Δ )_A^s$ |

CPAA

We show the existence of multiple solutions of a perturbed polyharmonic
elliptic problem at critical growth with Dirichlet boundary conditions when
the domain and the nonhomogenous term are invariant with respect to some
group of symmetries.

CPAA

We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic
problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
The semi-linear problems are studied in a framework where the associated functional is of
class $C^1$ but not of class $C^2$.

DCDS-S

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the $N$-Laplacian operator.

DCDS-S

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.

DCDS

We consider a class of scalar field equations with anisotropic nonlocal nonlinearities.
We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to
this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all
level below a certain threshold. We deduce the existence of a ground state when the variable
exponent slowly approaches the limit at infinity from below.

DCDS-B

We discuss the application of the Mountain Pass Algorithm to the
so-called quasi-linear Schrödinger equation,
which is naturally associated with a class of nonsmooth functionals so
that the classical algorithm cannot directly be used.
A change of variable allows us to deal with the lack of regularity.
We establish the convergence of
a mountain pass algorithm in this setting.
Some numerical experiments are also performed and lead to
some conjectures.

CPAA

We study the spectral structure of the complex linearized operator
for a class of nonlinear Schrödinger systems, obtaining as byproduct some
interesting properties of non-degenerate ground state of the associated elliptic
system, such as being isolated and orbitally stable.

CPAA

We study the combined effect of
concave and convex nonlinearities on the number of positive
solutions for a fractional system
involving critical Sobolev exponents. With the help of the
Nehari manifold, we prove that the system admits at least two
positive solutions when the pair of parameters $(\lambda,\mu)$
belongs to a suitable subset of $R^2$.

## Year of publication

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