Related papers: Dual Variational Methods for a nonlinear Helmholtz…

In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N},…

This article develops a duality principle applicable to a large class of variational problems. Firstly, we apply the results to a Ginzburg-Landau type model. In a second step, we develop another duality principle and related primal dual…

We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on }…

For $q \in (0, \infty)$, we consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_\xi f(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} in a…

We study the Schr\"{o}dinger-Poisson type system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left( \mu _{11}\phi _{u}-\mu _{12}\phi _{v}\right) u=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert u+e^{i\theta }v\right\vert…

A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order…

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1),…

The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^{\nu}} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^{\nu}}, \] where $p, q \in\mathbb{R}^+$ and $\nu\in…

In this work we are concerned with the following class of equations \[ -\Delta_p u -\lambda h(x)|u|^{p-2}u=f(x)|u|^{\gamma-2}u, \quad \mbox{in } \mathbb{R}^N, \] involving indefinite weight functions. The existence of solution may depend on…

In this article, we examine two double-phase variable exponent problems, each formulated within a distinct framework. The first problem is non-variational, as the nonlinear term may depend on the gradient of the solution. The first main…

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x) |v|^{q(x)-2}v -…

We are concerned with the following system of two coupled time-independent Gross-Pitaevskii equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~ \R^N,\\ -\Delta v+\lambda_2…

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2, \ \…

Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient…

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N}…

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-\Delta)^s v = \mu \quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure…

We are concerned with the existence of solution of the problem $ -\Delta ^H_pu+|u|^{p-2}u=\lambda|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quad\Omega,$ $u>0\quad \mbox{in}\quad\Omega,$ $a(\nabla u)\cdot \nu =0\quad \mbox{on}\quad\partial…

We prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ differential $k$-form with $\delta \left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k…

This article develops duality principles applicable to non-convex models in the calculus of variations. The results here developed are applied to Ginzburg-Landau type equations. For the first and second duality principles, through an…

In this paper, we study the following system \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u + V(x)u-(2\omega+\phi)\phi u=\lambda f(u)+|u|^{4}u, \ & \text{in} \ \mathbb{R}^{3}, \Delta \phi + \beta\Delta_4\phi = 4\pi(\omega+\phi)…