Related papers: Pathological solutions to elliptic problems in div…

We consider the problem $(P)$, $$ -\Delta u =c(x)u+\mu|\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega),$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $\mu>0, \, c \in…

In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -\Delta u = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} \Omega,\vdois\ -\Delta v = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} \Omega,\vdois\…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-\Delta)^{s} u-\mu\frac{u}{|x|^{2s}}-\lambda u=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} \Omega,\ \ \, u=0\,&\text{in} \R^N\backslash\Omega…

We consider the following elliptic system \Delta u =\nabla H (u) \ \ \text{in}\ \ \mathbf{R}^N, where $u:\mathbf{R}^N\to \mathbf{R}^m$ and $H\in C^2(\mathbf{R}^m)$, and prove, under various conditions on the nonlinearity $H$ that, at least…

We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann…

The aim of this paper is to prove the existence of multiple solutions for a family of nonlinear elliptic systems in divergence form coupled with a pointwise gradient constraint: \begin{align*} \left\{ \begin{array}{ll}…

\[ \left\{ \begin{array} [c]{lll} -\left( \Delta_{p}+\Delta_{q(p)}\right) u=\lambda_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})\delta_{x_{u}} & \mathrm{in} & \Omega\\ u=0 & \mathrm{on} & \partial\Omega, \end{array} \right. \] where…

In this paper, we establish optimal a priori $C^{1,\alpha}$ regularity estimates for the ratio $w = v/u$ of two solutions to the same elliptic equation $-\operatorname{div}(A \nabla u )=0$ with Lipschitz coefficients $A$, under the…

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+\lambda u=f+\mathrm{div } \mathbf{F}\quad \text{in }…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where $\Delta_p$ is the $p-$laplacian, with $1<p<n$. The…

Let $u$ be a bounded positive solution to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is a locally Lipschitz continuous function. Among other things, we show that if…

Let $p,q$ be functions on $\mathbb{R}^{N}$ satisfying $1\ll q\ll p\ll N$, we consider $p(x)$-Laplacian problems of the form \[ \left\{ \begin{array} [c]{l}% -\Delta_{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda\vert u\vert…

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give…

We consider model semilinear elliptic equations of the type \[ \begin{cases} - \mathrm{div} (A(x) \nabla u) = f u^{- \lambda}, \quad u > 0 \quad \text{in} \ \Omega, \\ u \in H_{0}^{1}(\Omega), \end{cases} \] where $\Omega$ is a bounded…

Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent…

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$,…

In this paper we deal with parabolic problems whose simplest model is $$ \begin{cases} u'- \Delta_{p} u + B\frac{|\nabla u|^p}{u} = 0 & \text{in} (0,T) \times \Omega,\newline u(0,x)= u_0 (x) &\text{in}\ \Omega, \newline u(t,x)=0 &\text{on}\…

We study a degenerate elliptic equation, proving existence results of distributional solutions in some borderline cases.

We consider positive solutions of the problem \begin{equation} \left\{\begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=0\qquad \mbox{in}\;\;\mathbb{R}_+^n,\\ \frac{\partial u}{\partial \nu^a}=u^{q} \qquad \mbox{on}\;\;\partial…