Related papers: Existence of solution for a class of indefinite va…

In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -\Delta u=\left(\lambda f(x)-\int_{\R^N}K(x,y)|u(y)|^{\gamma}dy\right)u,\quad \mbox{in} \quad \R^{N}, \\…

In this paper, we consider the Schr\"odinger type equation $-\Delta u+V(x)u=f(x,u)$ on the lattice graph $\mathbb{Z}^{N}$ with indefinite variational functional, where $-\Delta$ is the discrete Laplacian. Specifically, we assume that $V(x)$…

We look for a solutions to a nonlinear, fractional Schr\"odinger equation $$(-\Delta)^{\alpha / 2}u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where potential $V$ is coercive or $V=V_{per} + V_{loc}$ is a sum of periodic…

The purpose of this paper is to study $T$-periodic solutions to [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P) u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N where $s\in (0,1)$, $N>2s$, $T>0$, $m> 0$ and…

We consider a general elliptic equation $$ -\Delta_g u+V(x)u=f(x,u)+g(x,u^2)u $$ on a closed Riemannian manifold $(M, g)$ and utilize a recent variational approach by Hebey, Pacard, Pollack to show the existence of a nontrivial solution…

We study the following class of nonlinear Choquard equation, $$ -\Delta u +V(x)u =\Big( \frac{1}{|x|^\mu}\ast F(u)\Big)f(u) \quad \mbox{in} \quad \R^N, $$ where $0<\mu<N$, $N \geq 3$, $V$ is a continuous real function and $F$ is the…

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first…

In this article we are interested on the non-homogeneous fractional Schr\"odinger equation \begin{eqnarray}\label{eq00} &(-\Delta)^{\alpha}u(x) + V(x)u(x) = f(u) + h(x) \mbox{ in } \mathbb{R}^{n}. \end{eqnarray} By using mountain pass…

This paper is devoted to the following class of nonlinear fractional Schr\"odinger equations: \begin{equation*} (-\Delta)^{s} u + V(x)u = f(x,u) + \lambda g(x,u), \quad \text{in}\: \mathbb{R}^N, \end{equation*} where $s\in (0,1)$, $N>2s$,…

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N},…

In this paper, we establish the existence of one solution for a Schr\"{o}dinger equation with jumping nonlinearities: $-\Delta u+V(x)u=f(x,u)$, $x\in \mathbb {R}^N$, and $u(x)\to 0$, $|x|\to +\infty$, where $V$ is a potential function on…

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha…

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad…

Under simple hypotheses on the nonlinearity $f$, we consider the fractional harmonic operator problem \begin{equation}\label{abstr}\sqrt{-\Delta+|x|^2}\,u=f(x,u)\ \ \textrm{in }\ \mathbb{R}^N\end{equation} or, since we work in the extension…

We establish the existence of multi-bump solutions for the following class of quasilinear problems $$ - \Delta_{ p(x) } u + \big( \lambda V(x) + Z(x) \big) u ^{ p(x)-1 } = f(x,u) \text{ in } \mathbb R^N, \, u \ge 0 \text{ in } \mathbb R^N,…

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 $N \geq 2$,…

This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a…

In this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation:\[ \begin{cases} \frac{\partial u}{\partial t}= \Delta_{\mathbb{G},p}u+V(x)u^{p-1} &…

The main purpose of this paper is to study the existence of solutions for the following hybrid nonlinear fractional pantograph equation $$ \left\{\begin{aligned} &D_{0+}^\alpha…

We consider the problem {\Delta}u+V(x)u = f'(u) in RN. Here the nonlinearity has a double power behavior and V is invariant under an orthogonal involution, with V ({\infty}) = 0. An existence theorem of one pair of solutions which change…