In this paper, we study the fifth order Kadomtsev–Petviashvili II (KP–II) equation on the upper half–plane $U=\{(x,y)\in \mathbb R^2: y>0\}$. In particular, we obtain low regularity local well–posedness using the restricted norm method of Bourgain and the Fourier–Laplace method of solving initial and boundary value problems. Moreover, we prove that the nonlinear part of the solution is in a smoother space than the initial data.

We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $\alpha$-stable operator and the second one (possibly degenerate) corresponds to a class of lower order Lévy measures. Such operators do not have a global scaling property. We establish Hölder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.

In this paper, we consider nontrivial solutions of a quasilinear elliptic system with the weight functions $h(x)$ and $f_i(x)$ $(i = 1, 2)$ by applying the Nehari manifold method along with the fibrering maps and the minimization method. We analyze the effect of the parameters and weight functions on the existence and multiplicity of nontrivial solutions for the quasilinear elliptic system. When $(\lambda_1 f_1)^+ = (\lambda_2 f_2)^+ \equiv 0$, we prove that the system has at least one nontrivial nonnegative solution; and when $(\lambda_1 f_1)^+ \not \equiv 0$ or $(\lambda_2 f_2)^+ \not \equiv 0$, we show that the system has at least two nontrivial nonnegative solutions for any $(\lambda_1, \lambda_2)$ belonging to a certain subset of $\mathbb{R}^2$. These results are novel even for the corresponding scalar quasilinear elliptic equations and semilinear elliptic systems, which improve and extend the existing results in the literature.

In the paper, we establish the local well-posedness of the Schödinger-Boussinesq system on the half line with data of low regularity. The proof is based on the explicit solution formula of the linear boundary problem and the restricted norm method. Besides, we prove that the nonlinearity is smoother than the initial data. Our result match the known result on the full line in [9].