Related papers: A sign-changing solution for the Schr\"odinger-Poi…
In this paper, we study the non-homogeneous nonlinear Schr\"{o}dinger system $$\left\{ \begin{array}{ll} -\triangle u_j+V_j(x) u_j=g_j(x,u_1,\cdots,u_m)+h_j(x),& x\in \Omega,\\ \\ u_j:=u_j(x)=0,& x\in \partial\Omega,\\ \\ j=1,2,\cdots,m,…
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate…
In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schr\"{o}dinger-Poisson equations: $$-\Delta u+V(x)u+(|x|^{-1}*u^2)u-|u|^\frac{4}{3}u=\lambda u,~~x\in\R^3$$ on the…
A possible method to solve the sign problem is developed by modifying the original theory. Considering several modifications of the partition function, the observable in the original theory is reconstructed from the identity connecting the…
We give decay estimates of the solution to the linear Schr\"odinger equation in dimension $d \geq 3$ with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the…
We study the existence of sign changing solutions to the following problem $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} \Delta u+|u|^{p-1}u=0 \quad & {\rm in} \quad \Omega_\epsilon; u=0 \quad & {\rm on} \quad\partial \Omega_\epsilon,…
We show that a real eigenfunction of the Schr\"odinger operator changes sign near some point in $\mathbb{R}^n$ under a suitable assumption on the potential.
In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness…
We study the existence of $L^2$ normalized solutions for nonlinear Schr\"odinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional on $S_m=\{ u; \,…
We consider singular quasilinear elliptic systems with homogeneous Dirichlet boundary condition. Using Leray-Schauder topological degree, combined with the sub-supersolutions method and suitable truncation arguments, we establish the…
In this paper we are concerned with some $p$-Kirchhoff type problems involving sign-changing weight functions. We prove the existence of multiple positive solutions of the problem via the Nehari manifold approach.
In this paper, we study the concentration and multiplicity of solutions to the following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)+u^{2_s^{\ast}-1} &…
We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u + \psi u = \lambda |u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill -…
In this paper we study a Schr\"odinger-Bopp-Podolsky system of partial differential equations in a bounded and smooth domain of $\mathbb R^3$ with a non constant coupling factor. Under a compatibility condition on the boundary data we…
On compact Riemannian manifold of dimension n, and under some conditions on the curvature, we have changing-sign solutions for n large enough for an elliptic PDE.
Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schr\"odinger-Poisson system: $$ \left\{ \begin{array}{l} \vspace{2mm} -\Delta u+V(x)u+K(x)\phi(x)u=u^5 -\Delta…
In this paper, we investigate the existence of sign-changing solutions to a critical elliptic equation involving a Yamabe type operator on a compact manifold with boundary. The existence result is assured under some geometric conditions.
We study the following nonlinear Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u + \omega u + q^{2}\phi u = |u|^{p-2}u -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2 \end{cases} \hbox{ in }\mathbb{R}^3 \] with $a,\omega>0$. We…
In this paper, we study the following nonlinear Schr\"odinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=f(u)&\text{in}\ \mathbb{R}^3,\\ -\varepsilon^2\Delta \phi=K(x)u^2&\text{in}\…
In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3,…