相关论文: Existence of solutions for a semirelativistic Hart…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-\Delta+m^2)^\sigma u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<\sigma<1$, $V$ is a…

In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation $\sqrt{-\varepsilon^2 \Delta + m^2}u + V u = (I_\alpha * |u|^{p}) |u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V \colon…

We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -2} u \quad\text{in…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{-\Delta+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded…

We study the semilinear equation $-\Delta_g u + V(\sigma) u = f(u)$ on a Cartan-Hadamard manifold ${\cal M}$ of dimension $N \geq 3$, and we prove the existence of a nontrivial solution under suitable assumptions on the potential function…

We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -\Delta + m^2} \ u - mu + V(x)u - \frac{\mu}{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u…

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N}…

It is shown that exact solutions may be found for the energy eigenvalue problem generated by the class of semirelativistic Hamiltonians of the form H = sqrt{m^2+p^2} + hat{V}, where hat{V} is a non-local potential with a separable kernel of…

We study nondegeneracy of ground states of the Hartree equation $$ -\Delta u+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a…

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-\Delta + m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~\Omega, with the Dirichlet boundary condition $u=0$ on…

: We establish existence of an infinite family of exponentially-decaying non-radial $C^2$ solutions to the equation $\Delta u + f(u) = 0$ on $R^2$ for a large class of nonlinearities $f$. These solutions have the form $u(r,\theta )=e^{i…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying…

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem \[ \left\{ \begin{array} [c]{ll} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),~{\rm{in}}~ \mathbb{R}^{3},\\ u\in…

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension $3$. The Hartree type nonlinearity is $[V * |u|^2]u$ and the potential $V$ which generalizes the Yukawa has some growth condition.…

Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region…

In this paper the question of finding infinitely many solutions to the problem $-\Delta u+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…

This paper is concerned with the existence of solutions to the problem $$-\left(a+ b\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx \right)\Delta u +V(x)u+\lambda u = |u|^{p-2}u,\ \ x \in \mathbb{R}^{N},\ \ \lambda \in \mathbb{R}^{+} $$ where $a,…

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schr\"odinger equation $$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = |u|^{p-1}u\quad…