In this paper, we study a class of the critical Schrödinger-Bopp-Podolsky system with $p$-Laplacian in $\mathbb{R}^3$. Under different perturbation terms, the existence and multiplicity of nontrivial solutions are obtained by using several critical point theorems, respectively. Considering the influence of the $p$-Laplacian operator, critical and nonlocal terms, which cause the loss of the compactness condition, we attempt to overcome this difficulty by using the concentration compactness principle and some clever analysis. The results in this paper can be viewed as complementary to the previous results in the case of $p = 2$ and subcritical case.
The paper deals with the existence of normalized ground state solution to the following Kirchhoff equation \begin{equation} \begin{cases} \displaystyle -(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u +\lambda u=g(u)\quad in \ \mathbb{R}^{3},\\[6pt] u\in H^{1}(\mathbb{R}^{3}),\\ \displaystyle \int_{\mathbb{R}^{3}}|u|^{2}dx=\rho > 0 , \end{cases} \tag{0.1} \end{equation} where $ a, b > 0 $ are constants, $\rho$ is specified and $(\lambda,u)\in\mathbb{R}\times H^{1}(\mathbb{R}^{3})$ is determined. Under certain assumptions on $g$, we prove that (0.1) has the existence of least energy solutions to the normalized problem by combining variational methods with analytical skills.
After providing a concise derivation of a time-dependent model for the near-surface dynamics of the Antarctic Circumpolar Current with variable density, we present a variational formulation for stationary solutions as critical points of an appropriate functional, which is constructed from conservation laws for the system.
Starting from the general equations (in a rotating frame) for a compressible, viscous fluid, coupled to an equation of state and the first law of thermodynamics, we present a derivation based on the thin-shell approximation. This uses a single small parameter (epsilon), measuring the thinness of the shell, keeping all other parameters fixed as epsilon -> 0. The resulting equations retain the essentials of the spherical geometry, but are inviscid (at leading order) and allow a special structure in the vertical direction. Two exact solutions of this nonlinear system are described; one represents the Great Red Spot (GRS) in some detail, enabling the streamlines, the temperature distribution and the vorticity to be determined. The other solution is appropriate for the oscillatory (filamentary) structures that exist to the South (and East), and to the North (and West), of the GRS, although the details require the use of an approximation (based on elementary functions), resulting in a reasonably accurate analytical estimate of the solution, which is presented graphically. This oscillatory solution, however, does not possess an amplitude modulation in the azimuthal direction, but we may accommodate a modulation in the meridional direction. The solutions that we have obtained are compared with the available data, which includes the r & ocirc;le of heating associated with the GRS; nevertheless, the emphasis throughout is on the mathematical structure and properties of our system of equations. The significance of the work is discussed, and we also itemize the various issues that require further investigation.
