This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.

In this paper, we analyze main properties of the volume and layer potentials as well as the Poisson integral for a multi-dimensional degenerate parabolic equation. As consequences, we obtain trace formulae of the heat volume potential and the Poisson integral which solve Kac's problem for degenerate parabolic equations in cylindrical domains.

Recently, S. Kondo and A. Tani in SIAM J. Math. Anal. (see [9]) investigated the existence and uniqueness of the strong solution to the initial boundary value problem (IBVP) of electromagnetic fluid equations (1.4) with the magnetic-curvature-driven Rayleigh–Taylor instability on bounded domain in 3D. The present paper will improve and extend the results from bounded domain to $\mathbb{R}^3$. First, we establish the local well-posedness of the Cauchy problem for the equation (1.4) and obtain some important estimates of the solution to the plasma equations in $\mathbb{R}^3$ by some lemmas, thanks to these lemmas, we establish the global solution of the Cauchy problem of the equation. Secondly, the existence of global attractor of the plasma equations in a bounded domain of $\mathbb{R}^3$ is established. Finally, we obtain the Hausdorff and fractal dimensions of the global attractor of the equation.

We prove quasi-invariance of Gaussian measures $\mu_s$ with Cameron-Martin space $H^s$ under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order in dimension $d=2$ and sub-quintic nonlinearities in dimension $d=3$, for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov and Gunaratnam-Oh-Tzvetkov-Weber who proved this result for a cubic nonlinearity and $s$ an even integer. The main contributions are a modified construction of a weighted measure adapted to the higher order nonlinearity, and an energy estimate for the derivative of the energy replacing the integration by parts argument introduced in previous works. We also address the question of (non) quasi-invariance for the dispersionless model raised in the introductions to [15, 10].

The model system of equations which describes the self-propelled motion of point mass objects driven by camphor is a diffusion equation coupled with a system of nonlinear ordinary differential equations. If the objects have masses, then the motion of objects becomes very complicated when some of the objects hit the boundary of a water surface or collide each other. To avoid such complexity and try to get some general perspective for the motion, it is convenient to consider point mass objects, i.e., objects without areas. We give an existence of a weak solution of this model system by giving an a priori estimate for the solution. The key to this estimate is the choice of a special test function. This is a first step toward analyzing collective motion of point mass camphors. As far as we know, this result is the first result on the existence of a weak solution for this system.