In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,D_n)\}_{n=1}^\infty$, where $\mathcal{P} = \{p_n\}_{n=1}^\infty$ is a sequence of positive integers with $p_n > 1$ and $\mathcal{D} = \{D_n\}_{n=1}^\infty$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#D_n \in \{2,3,N_n\}$. We find a countable set $\Lambda \subset \mathbb{R}$ such that the set $\{e^{-2\pi i\lambda x}|\lambda \in \Lambda\}$ is an orthonormal basis of $L^2(\mu_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $\mu_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $\mu_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}$.
We study a class of Schrödinger-Poisson problems involving the nonlinearity $f(x)|u|^{p-2}u$ $(2 < p < 4)$ in $\mathbb{R}^3$. Such problems cannot be studied directly by applying the general Nehari manifold, because $(PS)$sequence may not be bounded. In this paper, by developing some useful analytical techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of equations, we resolve an open question posed in [S. Kim and J. Seok, Commun. Contemp. Math., 14 (2012)] of whether there exists a sign-changing solution $u_k^\lambda$changing signs exactly $k$ times to the problem above for the remaining range $2 < p < 4$. Furthermore, the energy of $u_k^\lambda$ is strictly increasing in $k$, as well as some asymptotic behaviors of $u_k^\lambda$ are obtained.
We study the one-dimensional Cauchy problem for a class of degenerate parabolic equations arising in superfast diffusion. For such equations, weak solutions are known to be non-unique. We establish the existence and uniqueness of semigroup solutions and prove the analyticity of the free boundary by means of the maximal regularity theory of Da Prato and Grisvard [14]. Finally, under suitable assumptions, we demonstrate the preservation of spatial convexity for an appropriate power of the solution.
This paper deals with minimizers for noncoercive integral functional of the type $$\mathcal{J}(v) = \int_{\Omega} a(x)|\nabla v(x)|^p dx - \int_{\Omega} f(x)v(x)dx, \quad v \in W_0^{1,p}(\Omega), $$ with $1 < p < n$, $0 < a(x) \leq \beta$, a.e. $\Omega$ and $\frac{1}{a(x)}$ and $f(x)$ belong to some Lebesgue or Marcinkiewicz spaces. It is shown by Weierstrass Theorem that such a functional has a minimizer in a larger space $W_0^{1,q}(\Omega)$ for an appropriate exponent $1 < q < p$. Some regularity properties are given by using Stampacchia Lemma. This paper also considers regularizing effect of an interplay between the coefficient of zero order term and the datum in noncoercive integral functional of the type $$\mathcal{I}(v) = \int_{\Omega} a(x)|\nabla v(x)|^p dx + \int_{\Omega} b(x)|v(x)|^p dx - \int_{\Omega} f(x)v(x)dx, \quad v \in W_0^{1,p}(\Omega). $$ It is shown that, even if $0 < b(x)$ and $f(x)$ belong only to $L^1(\Omega)$, the interplay \[ |f(x)| \leq 2Q b(x) \] implies the existence of a minimizer $u \in W_0^{1,q}(\Omega)$ satisfying $|u| \leq Q$.
In this paper, we study the 3D double-diffusive convection system. Using energy methods and Littlewood-Paley decomposition theory, we establish regularity criteria for the system in Lorentz spaces and sum spaces, respectively.
