We study the evolution equation associated with the biharmonic operator on infinite cylinders with bounded smooth cross-section subject to Dirichlet boundary conditions. The focus is on the asymptotic behavior and positivity properties of the solutions for large times. In particular, we derive the local eventual positivity of solutions. We furthermore prove the local eventual positivity of solutions to the biharmonic heat equation and its generalizations on Euclidean space. The main tools in our analysis are the Fourier transform and spectral methods.

In this paper, several new existence theorems on pairs of solutions are obtained for the fourth order boundary value problem ${u^{(4)}}(t) = h(t,u(t))$ subject to $u(0) = u(1) = u''(0) = u''(1) = 0$. Further, if $h(t,u) = \mu f(t,u)$, we prove that there exists $ \mu ^ * > 0$ such that there is no solution to the above boundary value problem for $\mu > {\mu ^ * }$, and that there are multiple solutions of the above boundary value problem for $0 < \mu < {\mu^ * }$.

In this paper, we are going to investigate the existence of semiclassical nodal solutions for the following Choquard equation: \begin{equation}\nonumber -\varepsilon^{2}\Delta u+V(x)u= (I_{\alpha} \ast|u|^p)|u|^{p-2}u, \ \ \ \ \text{in}\quad \mathbb R^N, \end{equation} where $N\ge 3$, $\alpha\in (0,N),$ and $I_{\alpha}$ is the Riesz potential. Under suitable assumption on the potential $V(x)$, for $p\in (2,\frac{N+\alpha}{N-2})$, the semiclassical nodal solutions are obtained if the parameter $\varepsilon$ is small enough. Meanwhile, we also prove that there is no least energy nodal solutions if $p < 2$. Moreover, the limit of the least energy nodal solutions of the equation is used to solve the case of $p=2$.

In this paper, we consider the existence of multi-Peak solutions to the nonlinear Chern-Simons-Schrödinger system $$ \begin{cases} -ihD_0\Psi-h^2(D_1D_1+D_2D_2)\Psi+V\Psi=|\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0= -\frac 12ih[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0= \frac 12ih[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1= -\frac12|\Psi|^2, \end{cases} \qquad\qquad (0.1) $$ where $p > 2$ and $V(x)$ is a non-radial potential. Our main result states that for every positive integer $k$, we can find $h_0>0$ such that for $0 < h < h_0$, problem (0.1) has a nontrivial static solution $(\Psi_h, A_0^h, A_1^h,A_2^h)$. Moreover, $\Psi_h$ has $k$ positive peaks, which tend to the local maximum point of $V(x)$ as $h\to 0^+$.

This paper is concerned with the Cauchy problem of the Klein-Gordon-Schrödinger (KGS) equations with a defocusing nonlinearity in three spatial dimensions. The global wellposedness at $H^2$-regularity level and the growth bounds for the corresponding Sobolev norm of the solutions are obtained by applying Koch-Tzvetkov type Strichartz estimates and modified energy, which removes the restriction of the smallness for the initial data in the previous literature and extends the exponential growth bounds to polynomial case.

We show the existence of positive solutions for a class of $p\&q$ problems given by $$ \begin{cases} -div\left(a\left(|\nabla u|^p\right) |\nabla u|^{p-2}\nabla u\right)+V(z)b \left(|u|^p\right)|u|^{p-2}u = f(u)\ \ \mbox{in $\mathbb{R}^N,$} \\ u \in D^{1,p}(\mathbb{R}^{N})\cap D^{1,q}(\mathbb{R}^{N}), \end{cases} $$ $N\geq 3$ and $1 < p\leq q < N$. The potential $V$ can vanish at infinity with exponential decay and $f$ is a function with subcritical growth of class $C^1$. We use Del Pino and Felmer's arguments [18] in order to overcome the lack of compactness.