In this paper, we prove the existence of the positive ground state solution to fractional spin-1 Bose-Einstein condensation(BEC) in one dimension.

In this paper, the upper critical fractional Choquard equation is considered. The interest in studying such equation comes from its strong connections with mathematical physics, for example, the fractional quantum mechanics, the Lévy random walk models, and the dynamics of pseudo-relativistic boson stars. Another strong motivation is from some mathematical difficulties such as the coexistence of two fractional diffusions, the lack of compactness, and the attendance of the upper critical exponent. To overcome these difficulties, some methods and techniques need to be combined and a radially symmetric solution is obtained.

In this paper, we study the existence of positive radial solutions for a prescribed mean curvature problem on an exterior domain. Based on $C^1$-regularity of solutions, which is closely related to the property of nonlinearity $f$ near $0,$ and by using the global bifurcation theory, we establish some existence results when $f$ is sublinear at $\infty$.

We are concerned with the following $p(x)$-Laplacian equations in $\mathbb{R}^N$ $$ -\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u= f(x,u)\ \ {\rm in }\; \mathbb{R}^N. $$ Based on a direct sum decomposition of a space, we investigate the multiplicity of solutions for the considered problem. The potential $V$ is allowed to be no coerciveness, and the primitive of the nonlinearity $f$ is of super-$p^+$ growth near infinity in $u$ and allowed to be sign-changing. Our assumptions are suitable and different from those studied previously.