Related papers: On positive solutions of p-Laplacian-type equation…

We characterize the existence of solutions to the quasilinear Riccati type equation \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + \sigma \quad \text{in} ~\Omega, \\ u&=&0 \quad…

Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let $q\in ]0,1[$, $\alpha\in L^{\infty}(\Omega)$, with $\alpha>0$, and $k\in {\bf N}$. Then, the problem…

In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{\lambda,\Lambda}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where $\mathcal{M}_{\lambda,\Lambda}^+$ and…

Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a smooth bounded domain, $a\in C(\bar{\Omega})$ a sign-changing function, and $0\leq q<1$. We investigate the Robin problem \[ \begin{cases} -\Delta u=a(x)u^{q} & \mbox{in $\Omega$},\\ u\geq0…

We study solutions to variable-coefficient elliptic equations of the form $-\D(A(x) \nabla u) = \kappa u$, $\kappa>0$, in an exterior domain $\Om\subset \Rn$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich's…

This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\& q$ Laplacian operator with indefinite weights $$\left(-\Delta_p\right)^{\alpha}u +…

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{ in $\Omega$}, \\ \quad u>0 &\text{ in $\Omega$}, \\ \quad u=0 & \text{ on $\partial \Omega$}, \end{cases} \] where $L(v)=-{\rm…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta +…

In this paper we study entire radial solutions for the quasilinear $p$-Laplace equation $\Delta_p u + k(x) f(u) = 0$ where $k$ is a radial positive weight and the nonlinearity behaves e.g. as $f(u)=u|u|^{q-2}-u|u|^{Q-2}$ with $q<Q$. In…

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

For the boundary value problem $$\left\{ \begin{array}{rcll} -\Delta_p u+u^{p-1}&=&|x|^{\alpha}u^{q-1}&\mbox{in }\Omega,\\ \frac{\displaystyle\partial u}{\displaystyle\partial{\bf n}}&=&0&\mbox{on }\partial \Omega, \end{array}\right. $$ in…

We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -\Delta_1 u +g(u)|Du|=h(u)f & \text{in $\Omega$,} \\ u=0 & \text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$,…

We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-\Delta)^m u = K(x) u^{\frac{n+2m}{n-2m}} ~~~~~~ in ~ \Omega \backslash \Lambda $$ near their singular set $\Lambda$, where $\Omega \subset…

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish…

We study the local properties of positive solutions of the equation $-\Delta u+ae^{bu}=m|\nabla u|^q$ in a punctured domain $\Omega\setminus\{0\}$ of $\bf R^2$ where $m,a,b$ are positive parameters and $q>1$. We study particularly the…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^\infty(\Omega)$ such that $\Delta^m \varphi\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(\Omega)$ we prove the…

We prove Liouville theorem for the equation $\Delta_m v + v^p + M |\nabla v|^{q}= 0$ in a domain $\Omega\subset\mathbb R^n$, with $M\in \mathbb{R}$ in the critical and subcritical case. As a natural extension of our recent work \cite{MWZ},…