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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 =…

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 consider the problem $-\Delta u+\lambda u=u^{p-1}$, where $u\in H^1_0(\Omega)$ verifies $\|u\|_{L^2}=m>0$, and $\lambda\in [0,+\infty)$. Here, $\mathbb{R}^N\setminus\Omega$ is nonempty and compact. We prove the existence of a solution…

In this paper, we explore the bifurcation phenomena and establish the existence of multiple solutions for the nonlocal subelliptic Brezis-Nirenberg problem: \begin{equation*} \begin{cases} (-\Delta_{\mathbb{G}})^s u= |u|^{2_s^*-2}u+\lambda…

We study the homogeneous Dirichlet problem for the equation \[ u_t-\operatorname{div}\left((a(z)\vert \nabla u\vert ^{p(z)-2}+b(z)\vert \nabla u\vert ^{q(z)-2})\nabla u\right)=f\quad \text{in $Q_T=\Omega\times (0,T)$}, \] where…

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= \lambda w(x)|u|^{s-2}u+\theta B\left(x,u\right) \quad…

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…

We consider a second order nonlinear ordinary differential equation of the form $u'' + f(u) = p(t)$ where the forcing term $p(t)$ is a $T$-periodic function and the nonlinearity $f(u)$ satisfies the properties of Ambrosetti-Prodi problems.…

We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-\Delta u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded domain $\Omega \subset \R^N$, which…

We study the degenerate elliptic equation $-\mathop{\rm div}(|x|^\alpha\nabla u) =f(u)+t\phi(x)+h(x)$ in a bounded open set $\Omega$ with homogeneous Neumann boundary condition, where $\alpha\in(0,2)$ and $f$ has a linear growth. The main…

We investigate the existence and multiplicity of solutions to the following $p(x)$-Laplacian problem in $\mathbb{R}^{N}$ via critical point theory \begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup _{p(x)}u+V(x)\left\vert…

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,~~~~~…

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 $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega $ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem…

In this paper we consider a class of critical concave convex Ambrosetti-Prodi type problems for the fractional $p$-Laplacian operator. By applying the Linking Theorem and the Mountain Pass Theorem as well, the interaction of the…

This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=|u|^4u+\lambda|u|^{q-2}u\ \ &\mbox{in}\ \Omega, \displaystyle u=0\ \…

We investigate the eigenvalue problem $-\text{div}(\sigma \nabla u) = \lambda u\ (\mathscr{P})$ in a 2D domain $\Omega$ divided into two regions $\Omega_{\pm}$. We are interested in situations where $\sigma$ takes positive values on…

We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega,…

Global self-similar solutions to the parabolic Hardy-H\'enon equation $$ u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $\sigma>\max\{-2,-N\}$. The…

We deal with Dirichlet problems of the form $$ \Delta u+f(u)=0 \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial \Omega $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 3$, and $f$ has supercritical growth from the viewpoint…