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In this paper, we study the following fully nonlinear elliptic equations \begin{equation*} \left\{\begin{array}{rl} \left(S_{k}(D^{2}u)\right)^{\frac1k}=\lambda f(-u) & in\quad\Omega \\ u=0 & on\quad \partial\Omega\\ \end{array} \right.…

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schr\"odinger equation $$ \left\{ \begin{array}{ll} -\Delta u = \lambda u + f(u)\quad & \text{in…

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ \Delta^m u = \pm u^{\alpha} \quad \text{ in }…

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b…

We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuous function,…

This paper concernes with the existence of heteroclinic solutions for the following class of elliptic equations $$ -\Delta{u}+A(\epsilon x, y)V'(u)=0, \quad \mbox{in} \quad \Omega, $$ where $\epsilon >0$, $\Omega=\R \times \D$ is an…

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schr\"odinger (NLS) equation, namely, $i u_t - \Delta^2 u - 2a \Delta u + |u|^{\alpha} u = 0, ~ x,a \in \R$, $\alpha>0$, and investigate the dynamics of its solutions for…

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth…

In this paper, we study the asymptotic behavior of least energy nodal solutions $u_p(x)$ to the following fourth-order elliptic problem \[ \begin{cases} \Delta^2 u =|u|^{p-1}u \quad &\hbox{in}\;\Omega, \\ u=\frac{\partial u}{\partial \nu}=0…

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 deal with the planar Schr\"{o}dinger-Poisson system \begin{equation*}\begin{cases} -\Delta u + V(x) u + \phi u = b|u|^{p-2} u \ &\text{in}\ \mathbb{R}^{2},\\\Delta \phi= u^{2} &\text{in}\ \mathbb{R}^{2},\end{cases}…

This paper analyzes the structure of the set of nodal solutions of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite astonishingly, the…

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u + \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\ \partial\Omega,…

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 a new class of fully nonlinear uniformly elliptic equations with a so-called harmonic map-like structure, whose model case is given by \begin{equation*} \mathcal{M}^{\pm}_{\lambda,\Lambda}(D^2u) \pm b(x) |Du| \pm…

In this work, we study the following Kirchhoff equation $$\begin{cases}-\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)\Delta u +u =Q(x)u^{q-1},\quad u>0,\quad x\in {\mathbb{R}^{3}},\\u\to 0,\quad \text{as}\ |x|\to…

Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type $ -\Delta u + u= a(x) f(u)$ in $ \Omega$ with…

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_\Omega |\nabla u|^2 \dif x +b\right)\Delta u = \lambda u+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $\lambda \in…