This paper investigates a class of PDEs with coefficients in negative Besov spaces and whose solutions have linear growth. We show existence and uniqueness of mild and weak solutions, which are equivalent in this setting, and several continuity results. To this aim, we introduce ad-hoc Besov-H\"older type spaces that allow for linear growth, and investigate the action of the heat semigroup on them. We conclude the paper by introducing a special subclass of these spaces which has the useful property to be separable.

In this paper, we study theEinstein-scalar field Lichnerowiczequations on CR manifolds in the null case.We prove some existence resultsinvolving the equations.

In this paper, we consider the existence of stationary solutions to a class of quasilinear Schrödinger type equations. As a main novelty with respect to previous results, we are able to deal with locally super-linear nonlinearities in a fairly general framework and prove the existence of finite energy nodal solutions. The proof is accomplished by a new variational perturbation approach together with a delicate analysis of the asymptotic behavior of descending flow.

This paper is concerned with a nonlocal critical Neumann problem in the half space. By virtue of variational methods, the global compactness theorem and Brouwer theory of degree, we obtain the existence and multiplicity of positive solutions.

In this paper, we consider the linearized compressible Navier-Stokes equations in the whole space $\mathbb{R}^n$. Concerning initial datum with suitable regularities, we introduce a new threshold $|\mathbb{B}_0|=0$ to distinguish different large-time behaviors. Particularly in the lower-dimensions, optimal growth estimates ($n=1$ polynomial growth, $n=2$ logarithmic growth) hold when$|\mathbb{B}_0| > 0$, whereas optimal decay estimates hold when $|\mathbb{B}_0|=0$. Furthermore, we derive asymptotic profiles of solutions with weighted $L^1$ datum as large-time.

It is shown by means of reiterated two-scale convergence in the Sobolev-Orlicz setting that thesequence of solutions of a class of highly oscillatory problems involvingnonlinear elliptic operators with nonstandard growth, convergesto a solution of a suitable homogeneous nonlinear elliptic equation associated to an operator with nonstandard growth.