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We study entire bounded solutions to the equation $\Delta u - u + u^3 = 0$ in $\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a…

In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive,…

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic…

We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above…

Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in…

We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither…

In this paper, we study important Schr\"{o}dinger systems with linear and nonlinear couplings \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1-\lambda_1 u_1=\mu_1 |u_1|^{p_1-2}u_1+r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+\kappa…

We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an…

We show that ground state solutions to the nonlinear, fractional problem \begin{align*} \left\{ \begin{array}{ll} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ \Omega, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus…

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

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schr\"odinger equations \begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

We construct a new family of entire solutions for the nonlinear Schr\"odinger equation \begin{align*} \begin{cases} -\Delta u+ V(y ) u = u^p, \quad u>0, \quad \text{in}~ \mathbb{R}^N, \\[2mm] u \in H^1(\mathbb{R}^N), \end{cases}…

For the Schr\"odinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and…

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: {displaymath} {cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega, -\Delta u_2 +\la_2 u_2…

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schr\"{o}dinger equation $$iu_t-\triangle u +|u|^2u+\frac{\partial{f(x,u,\bar u)}}{\partial{\bar u}}=0, \quad…

We consider the following nonlinear problem in $\R^N$ $$\label{eq} - \Delta u +V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that if $V(r)$ has the following…

We are concerned with $T$-periodic solutions of nonautonomous parabolic problem of the form $u_t = \Delta u + V(x) u + f(t,x,u)$, $t >0$, $x \in \mathbb{R}^N$, with $V \in L^\infty (\mathbb{R}^N)+L^p(\mathbb{R}^N)$, $p \geq N$ and…

By employing a novel perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the…

In this paper we consider the following problem $$\begin{cases} -\Delta_{g}u+V(x)u=\lambda\alpha(x)f(u), & \mbox{in }M\\ u\geq0, & \mbox{in }M\\ u\to0, & \mbox{as }d_{g}(x_{0},x)\to\infty \end{cases}$$where $(M,g)$ is a $N$-dimensional…