We study the nonlinear Schrödinger equation with initial data in $ \mathcal Z ^s_p(\mathbb R^d)=\dot{H}^s(\mathbb R^d)\cap L^p(\mathbb R^d)$, where $0 < s < \min\{d/2,1\}$ and $2 < p < 2d/(d-2s)$. After showing that the linear Schrödinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.

In the present paper, the variational and the dual methods were used to show the existence of solutions to the following problems \begin{equation*} L_Ku=\rho(x)|u|^{p-1}u, \ \mbox{in}\ \mathbb R^N, \tag*{(P)} \end{equation*} and \begin{equation*} L_Ku=\varepsilon a(x)|u|^{p-1}u+\rho(x)|u|^{q-1}u, \ \mbox{in}\ \mathbb R^N, \tag*{(Q)} \end{equation*} where $0 < p < 1 < q$, $a \in L^{\frac{q+1}{q-p}}(\mathbb R^N)$, $\rho(x)$ is a continuous positive function, $\varepsilon>0$ and $L_K$ is a non-local dispersion operator.

In this work we prove the existence of an autonomous Hamiltonian vector field in $W^{1,r}(\mathbb T^d; \mathbb R^d)$ with $r < d-1$ and $d \geq 4$ for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio's superposition principle [2], we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover, we show that the Hamiltonian is not constant along these integral curves.

Global solution curve and exact multiplicity of positive solutions for a class of semilinear systems are obtained. The results extend some known global bifurcation results of P. Korman (2002) by allowing the nonlinearity to possess a singularity.Applications to fourth-order problems arising from MEMS/NEMS models give new global bifurcation results.

In this paper, we dedicate to the following Kirchhoff type equation with critical exponential growth \begin{equation*} -\varepsilon^2M \Big ( \int_{\mathbb R^2}|\nabla u|^2\mathrm{d} x \Big ) \Delta u+V(x)u=f(u),\ \ x\in\mathbb R^2, \end{equation*}where $\varepsilon>0$ is a parameter, $M\in\mathcal{C}(\mathbb R^+,\mathbb R^+), V\in\mathcal{C}(\mathbb R^2,\mathbb R)$ and $f\in\mathcal{C}(\mathbb R,\mathbb R)$. Under some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of $V$ in the semi-classical limit by using a homeomorphism between the ground states of Kirchhoff equation and a related semilinear local elliptic equation combined with variational methods.

Erratum for the article [1]: D. Cruz-Uribe and V. Naibo, Kato-Ponce inequalities on weighted and variable Lebesgue spaces, Differential Integral Equations, 29 (2016), no. 9/10, 801-836.