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In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&\Delta^2u-a_0\Delta u+(\lambda b(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\lambda})}% $$ where $N\geq3$,…

We study semilinear damped wave equations with power nonlinearity $|u|^p$ and initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\gamma}$. In the present paper, we obtain a new critical exponent…

We consider the sinh-Poisson equation $$(P)_\lambda\quad -\Delta u=\la\sinh u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\rr^2$ and $\lambda$ is a small positive parameter. If…

In this paper we investigate the existence of solutions to a nonlinear elliptic problem involving critical Sobolev exponent for a polyharmonic operator on a Riemannian manifold $M$. We first show that the best constant of the Sobolev…

We study the pure Neumann Lane-Emden problem in a bounded domain \[ -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on }\partial \Omega, \] in the subcritical, critical, and supercritical regimes. We show existence…

We prove the nonexistence of smooth stable solution to the biharmonic problem $\Delta^2 u= u^p$, $u>0$ in $\R^N$ for $1 < p < \infty$ and $N < 2(1 + x_0)$, where $x_0$ is the largest root of the following equation: $$x^4 -…

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. $$ has…

Let $(\mathcal{M},g)$ be a smooth compact Riemannian manifold of dimension $N\geq 8$. We are concerned with the following elliptic system \begin{align*} \left\{ \begin{array}{ll} -\Delta_g u+h(x)u=v^{p-\alpha \varepsilon}, \ \ &\mbox{in}\…

This paper deals with the asymptotic behavior as $t\rightarrow T<\infty$ of all weak (energy) solutions of a class of equations with the following model representative: \begin{equation*} (|u|^{p-1}u)_t-\Delta_p(u)+b(t,x)|u|^{\lambda-1}u=0…

In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This…

We consider the semilinear elliptic boundary value problem \[ -\Delta u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on }\partial\Omega, \] in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for supercritical…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

We study the regularity properties of a weak solution to the boundary value problem for the equation $-\Delta \rho +a u=f$ in a bounded domain $\Omega\subset \mathbb{R}^N$, where $\rho=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla…

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of \cite{DPFBS}, the existence of at…

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this…

We consider the equation $$-\Delta u+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the…

In this paper, we study the subcritical biharmonic equation \[\Delta ^2 u=u^\alpha\] on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature. Using the method of invariant tensors, we derive…

In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -\epsilon^2\Delta u +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -\epsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u &&\text{in }…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…