相关论文: Ground state solutions for a nonlocal equation in …

We study the following singularly perturbed nonlocal Schr\"{o}dinger equation $$ -\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u) \quad \mbox{in} \quad \R^2, $$ where $V(x)$ is a continuous real function on…

We study the existence of solution for the following class of nonlocal problem, $$ -\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \mbox{in} \quad \mathbb{R}^2, $$ where $V$ is a positive periodic potential,…

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}\Delta u+\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-2}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where…

We study the existence of solutions of the following nonlinear Schr\"odinger equation $$ -\Delta u+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic…

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

In this paper we are concerned with the fractional Schr\"{o}dinger equation $(-\Delta)^{\alpha} u+V(x)u =f(x, u)$, $x\in \rn$, where $f$ is superlinear, subcritical growth and $u\mapsto\frac{f(x, u)}{\vert u\vert}$ is nondecreasing. When…

In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u+V(x)u= f(u) \mbox{ in } \mathbb{R}^{N}, \end{align*} where $s\in (0,1)$, $N>…

We study the existence of solutions of the following nonlinear Schr\"odinger equation \begin{equation*} -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where…

We obtain the existence of ground state solution for the nonlocal problem $$ m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-\Delta u + b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a…

In this paper we study the existence of ground state solution for an indefinite variational problem of the type $$ \left\{\begin{array}{l} -\Delta u+(V(x)-W(x))u=f(x,u) \quad \mbox{in} \quad \R^{N}, u\in H^{1}(\R^{N}), \end{array}\right.…

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schr\"odinger} equations involving Stein-Weiss type convolution \begin{align*} -\Delta_N u -\Delta_N (u^{2})u +V(x)|u|^{N-2}u=…

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 are concerned with singular elliptic equations of the form $-\Delta u= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with…

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -\Delta u+V(x)u=A(\epsilon x)f(u), \quad x \in \R^{N}, \eqno{(P)_{\epsilon}} $$ where $N \geq 2$,…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…

We establish the existence of a positive ground state solution for a Kirchhoff problem in $\mathbb{R}^2$ involving critical exponential growth, that is, the nonlinearity behaves like $\exp(\alpha_{0}s^{2})$ as $|s| \to \infty$, for some…

In this paper, we study the following class of weighted Choquard equations \begin{align*} -\Delta u =\lambda u + \Bigg(\displaystyle\int\limits_\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ \Omega~~ \text{and}~~…

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger equation $(-\Delta )^s u + V(x) u = \lambda a(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0<s<\min\{1,N/2\},$…

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -\Delta u+V(x)u=A(\epsilon x)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in…

In this paper, we study the nonlinear Schr\"{o}dinger equation $ -\Delta u+V(x)u=f(x,u) $on the lattice graph $ \mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$…