Related papers: Multiple sign-changing and semi-nodal solutions fo…

In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$,…

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in…

This paper is concerned with the following logarithmic Schr\"{o}dinger system: $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta u_2+\omega_2u_2=\mu_2 u_2\log…

In this paper we establish the existence of bound state solutions of any given order to $$ \Delta_m u +f(u)=0, x\in R^N, N\ge m>1, (P) $$ where $\Delta_m u=\nabla\cdot(|\nabla u|^{m-2}\nabla u)$ using the same techniques as in [GST] to…

We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schr\"{o}dinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order…

In this paper we construct a new kind of positive solutions of $$\De u-u+u^{p}=0 \text{on} \R^2$$ when $p> 2.$ These solutions $\displaystyle{u(x,z)\sim \om(x-f(z))+ \sum_{i=1}^{\infty}\om_{0}((x, z)-\xi_i\vec{e}_{1})}$ as $L\rightarrow…

We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 =…

We show that the number of solutions of a double singularly perturbed Schroedinger Maxwell system on a smooth bounded domain A depends on the topological properties of the domain. In particular if A is non contractible we obtain cat(A) + 1…

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$…

We study the fractional Schr\"odinger equations with a vanishing parameter: $$ (-\Delta)^s u+u =|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $2<q<p\leq…

In the present paper we consider the coupled system of nonlinear Schr\"{o}dinger equations with the fractional Laplacian \[ \left\{ \begin{aligned} (-\Delta)^\alpha u_1 & = \lambda_1u_1+f_1(u_1)+\partial_1F(u_1,u_2)\ \ \mathrm{in}\…

We study a class of critical Schr\"odinger-Poisson system of the form \begin{equation*} \begin{cases} -\Delta u+\lambda V(x)u+\phi u=\mu |u|^{p-2}u+|u|^{4}u& \quad x\in \mathbb{R}^3,\\ -\Delta \phi=u^2&\quad x\in \mathbb{R}^3,\\ \end{cases}…

In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u+\tau|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4,…

We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or H\'{e}non term: $$-\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}(e^u-e^{-u}) \quad\mathrm{in}\quad \Omega,$$ $$\frac{\partial u}{\partial…

Let $\Omega\subset \mathbb{R}^{N}$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schr\"{o}dinger system with $d\geq 2$ equations \begin{equation*} -\Delta…

We study the existence and nonexistence of normalized solutions $(u_a, \lambda_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schr\"{o}dinger equation with mixed nonlocal nonlinearities. This study can be viewed as a…

We consider the following quasilinear Schr\"{o}dinger equations of the form \begin{equation*} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \mbox{in}\ \mathbb{R}^N\ \mbox{and}\ \underset{|x|\rightarrow \infty}{\lim} u(x)=0,…

In this paper, we study the following two-component systems of nonlinear Schr\"odinger equations \begin{equation*} \left\{\aligned&\Delta u-(\lambda a(x)+a_0(x))u+\mu_1u^3+\beta v^2u=0\quad&\text{in }\bbr^3,\\ &\Delta v-(\lambda…

In this paper, we study radial solutions of $\Delta u + K(|x|) f(u)+\frac{ (N-2)^2 u}{|x|^{2+(N-2)\delta}} =0, \ 0<\delta<2$ in the exterior of the ball of radius $R>0$ in ${\mathbb R}^{N}$ where $f$ grows superlinearly at infinity and is…

In this note we prove that: \begin{theorem} for $2\leq s<\frac{n}{2}$ or $1\leq s<\frac{2n}{n+1}$ or $1\leq s<\frac{n}{2}$ but n is even, $(-\Delta)^{s}(u)=|u|^{q-2}u,q=\frac{2n}{n-2s}$ has infinitely many sign changing solutions or…