Related papers: Fully nontrivial solutions to elliptic systems wit…

We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -\Delta u = \lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1} \] where $1 \le p<q<2$, $\lambda_+ >0$, $\lambda_- \ge 0$. The equation is…

Starting from a bound state (positive or sign-changing) solution to $$ -\Delta \omega_m =|\omega_m|^{p-1} \omega_m -\omega_m \ \ \mbox{in}\ \R^n, \ \omega_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ \Delta u_0 + \lambda…

We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation $$ -\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \qquad (P_\varepsilon)$$…

In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^\eta u =|x|^\eta f(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\,…

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -\Delta u + \lambda_1 u = u^3 + \beta uv^2, \ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v \ \text{in } \Omega,\qquad \frac{\partial…

We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \]…

In this paper, we consider the elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u=g(x,v)\,\, \textnormal{in}\Omega, & \hbox{} -\Delta v=f(x,u)\,\,\textnormal{in}\Omega, & \hbox{} u=v=0\textnormal{on}\partial\Omega, &…

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 paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…

We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous…

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -\Delta u +…

In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} \Delta^2u= \lambda u+\mu u\ln u^2+|u|^{2^{**}-2}u, &x\in\Omega,\\ u=\dfrac{\partial u}{\partial \nu}=0, &x\in\partial\Omega \end{cases}…

Via a constrained minimization, we find a solution $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = \rho \end{cases}…

We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schr\"odinger system \begin{align*} - \Delta u + \lambda_1 u &= \frac{\alpha u(\alpha…

We deal with the following semilinear equation in exterior domains \[-\Delta u + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. Assuming that the weight $a$ is positive and…

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm…

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ \left\{…

In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem \begin{equation} Lu=|u|^{p-2}u+\mu|u|^{q-2}u~~\text{in}~~\Omega,~~~~~…