We construct blowing-up solutions to Bopp-Podolsky-Schrödinger-Proca and Schrödinger-Poisson-Proca systems in the case of electro-magneto-static solutions when the background space is the standard $3$-sphere. The solutions exhibit multi-peaks.

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [7, 21] who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.

In this paper, we deal with a family of the fractional elliptic operators with difference in the Orlicz-Sobolev spaces under boundary and initial conditions. We get a theorem which shows existence of a sequence of weak solutions for the fractional elliptic problems with difference in the Orlicz-Sobolev spaces. We first show that there exists a sequence of weak solutions for this problem on the finite-dimensional subspace. We next show that there exists a limit sequence of the sequence of weak solutions for finite-dimensional problem and this limit sequence is the sequence of the solutions of our problem. We get this result by the estimate of the energy functional and the compactness property of the continuous embedding inclusions between some special spaces.