We consider the uniqueness of positive solutions to \begin{equation*} \begin{cases} \triangle u +f(u)=0 & \text{in $\mathbb{R}^n$},\\ \displaystyle \lim_{\lvert x \rvert \to \infty} u(x) =0, \end{cases} \end{equation*} where $f(u)=- \omega u+u^p-u^{2p-1},$ with $\omega>0$ and $p>1$. It is known that for fixed $p>1$, a positive solution exists if and only if $\omega < \omega_p:=\dfrac{p}{(p+1)^2}$. We deduce the uniqueness in the case where $\omega$ is close to $\omega_p$, from the argument in the classical paper by Peletier and Serrin [9], thereby recovering a part of the uniqueness result of Ouyang and Shi [8] for all $\omega \in (0, \omega_p)$. In the appendix we consider the more general nonlinearity \begin{equation*} f(u)=-\omega u + u^p - u^q, \qquad \omega>0, ~~q>p>1 \end{equation*} and discuss the existence and uniqueness conditions. There we prove the fact that $f$ having positive part is equivalent to $\tilde{f}$ remaining negative, where $ \tilde{f}(u):= (uf'(u))'f(u)-uf'(u)^2 . $

In this paper, we consider the Schrödinger equation with quadratic potential \begin{equation*} i\frac{\partial }{\partial t}u =-\triangle u+ \left \vert x \right \vert ^{2}u \text{ }in\text{ }\mathbb{R}^{n+1}, \text{ }u(x,0)=f(x)\in L^{2}(\mathbb{R} ^{n}). \end{equation*} Using Hermite functions and some other classical tools, we give an elementary proof of the Kato-type smoothing estimate: for $i\neq j\neq k,$ $ \delta \in \lbrack 0,1],$ and $n\geqslant 3,$ \begin{equation*} \int _{0}^{2\pi } \int _{\mathbb{R}^{n}}\frac{ \left \vert u(x,t) \right \vert ^{2}}{ \left ( x_{i}^{2}+x_{j}^{2}+x_{k}^{2} \right ) ^{\delta }} dxdt\leqslant C \left \Vert f \right \Vert _{2}^{2}. \end{equation*} This is equivalent to proving a uniform $L^{2}(\mathbb{R}^{n})$ boundedness result for a family of singularized Hermite projection kernels. As an application of the above estimate, we also prove the $\mathbb{R}^{9}$ collapsing variable-type Strichartz estimate \begin{equation*} \int _{0}^{2\pi } \int _{\mathbb{R}^{3}} \left \vert u(\mathbf{x}, \mathbf{x},\mathbf{x},t) \right \vert ^{2}d\mathbf{x}dt\leqslant C \left \Vert (-\triangle + \left \vert x \right \vert ^{2})f \right \Vert _{2}^{2} \ \ \ \text{ where $\mathbf{x\in }\mathbb{R}^{3}$.} \end{equation*}

The long-time behavior in the sense of distributions for stochastic Klein--Gordon--Schrödinger equations in the whole space ${\mathbb R}^n$, $1\leq n\leq 3$, is studied. First the existence of one stationary measure from any moment-finite initial data in the space $H^1({\mathbb R}^n)\times H^1({\mathbb R}^n) \times L^2({\mathbb R}^n)$ is proved and then a global measure attractor is constructed in the space consisting of probability measures supported on $H^2({\mathbb R}^n)\times H^2({\mathbb R}^n)\times H^1({\mathbb R}^n)$. Because of the lack of compact embedding, some a priori estimates and a split of solutions play important roles in the approach.

This paper is concerned with the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of the evolution of surface water waves. We prove a mass concentration property for $H^{1}$-solutions, analogous to the one known for the $L^{2}$-critical nonlinear Schrödinger equation. We also prove a mass concentration result for $L^{2}$-solutions.

We study existence and concentration of positive solutions for systems of the form $-\epsilon^2\Delta u + V(x)u = \nabla F(u)$ in $\mathbb{R}^N$ where $\epsilon \gt 0$ is a small parameter, $F : [0,\infty)^k \rightarrow \mathbb{R}$ is a $C_{loc}^{1,\alpha}$ function, $N \geq 3, k \geq 1$, and the potential $V$ has a positive infimum and a well.

We study the existence and multiplicity of positive solutions to $n\times n$ systems of the form \begin{align*} -\Delta u_1&=\lambda f_1(u_2)& \mbox{ in }\Omega\\ -\Delta u_2&=\lambda f_2(u_3)&\mbox{ in }\Omega\\ \vdots \quad &= \quad\vdots&\\ -\Delta u_{n-1}&=\lambda f_{n-1}(u_n)&\mbox{ in }\Omega\ \ -\Delta u_n&=\lambda f_n(u_1)&\mbox{ in }\Omega\\ u_1&=u_2=...=u_n=0 &\mbox{ on }\partial\Omega. \end{align*} Here $\Delta$ is the Laplacian operator, $ \lambda$ is a non-negative parameter, $\Omega$ is a bounded domain in $\mathbb R^N$ with smooth boundary $\partial\Omega$ and $f_i\in C^1([0,\infty)),$ $i\in\{1,2,\dots,n\},$ belongs to a class of strictly increasing functions that have a combined sublinear effect at $\infty$. We establish results for positone systems ($f_i(0)\geq0,$ $ i\in\{1,\dots,l-1,l+1,\dots,n\}$ and $f_l(0)>0$ for some $l\in\{1,\dots,n\}$), semipositone systems (no sign conditions on $f_i(0)$) and for systems with $f_i(0)=0,$ $ i\in\{1,2,\dots,n\}.$ We establish our results by the method of sub and supersolutions.

We address the well posedness of the Cauchy problem and the stability of solitary waves for a Boussinesq system in $\mathbb{R}^{1+2}$. We exploit the fact that this 2D system has a ``KdV'' structure in the sense that it takes the form $U_t =\mathcal A_0U+ \mathcal{\mathcal A}(U) U$, where $\mathcal A_0$ is a third-order linear operator and the entries of the operator $\mathcal{A}(U)(U)$ are linear combinations of products of powers of components of $U$ with its order one spatial derivatives, as in the well-known 1D-KdV model. Using this ``2D-KdV'' structure, we establish existence and uniqueness for the Cauchy problem associated with the Boussinesq type system by following Kato's approach for the generalized KdV equation. By a variational argument, we obtain global well posedness in time for small initial data. We prove orbital stability of solitary waves directly, by using a variational approach involving the characterization of the ground state solutions, as is done for some 2-D models.

We study the existence of positive solutions to classes of $n\times n$ systems of reaction diffusion equations with Dirichlet boundary conditions. In particular, we consider systems where the reaction terms approach $-\infty$ near the origin and satisfy a combined sublinear condition at $\infty .$ We use the method of sub-super solutions to establish our results.

This paper is concerned with the nonlinear partial differential equations of calculus of variations perturbed by noise in the Gelfand triple $V\subset H\subset V^{\prime }$. The main result is a Trotter-type theorem for this equation. In the second part of the paper we prove that, if we assume graph convergence of the sequence of nonlinear operators $ \{ A^{\alpha} \} _{\alpha }$, we have convergence of the corresponding sequence of invariant measures. Those results are used in the last part of the paper to study the homogenization problem for the equation, in the case of the differential operator of the type $A ( u ) =- \text{\mathrm {div}} [ a ( \nabla u ) ], $ for $u\in H_{0}^{1} ( \mathcal{O} ).$

We consider the Dirichlet problem $ (*)$ $-\Delta u = \mu u + f $ in $\Omega$, $u=0$ on $\partial \Omega$, with $\Omega$ either a bounded smooth convex domain in $\mathbb R^2$, or a ball or an annulus in $\mathbb R^N$. Let $\lambda_2$ be the second eigenvalue, with $\varphi_2$ an associated eigenfunction. Although the two nodal domains of $\varphi_2$ do not satisfy the interior ball condition, we are able to prove under suitable assumptions that, if $\mu$ is sufficiently close to $\lambda_2$, then the solution $u$ of $(*)$ also has two nodal domains which appear as small deformations of the nodal domains of $\varphi_2$. For $N=2$, use is made in the proof of several results relative to the Payne conjecture.