We consider the drifting p-Laplace operator

\begin{equation*}\Delta_{p,v}u=e^{-v} \text{div}\,(e^v |\nabla u|^{p-2}\nabla u)\end{equation*}

and discuss generalized weighted Hardy-type inequalities associated with the measure $d\mu=e^{v(x)}dx$. As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term

\begin{equation*}-\Delta_{p,v} u\ge c(x) u^{p-1}+B \frac{|\nabla u|^p}{u} \quad \text{in}\ \Omega,\end{equation*}

where Ω is a bounded or an unbounded exterior domain in ${\mathbb{R}}^N$, $N \gt p \gt 1$, $B+p-1 \gt 0$, as well as of the non-autonomous quasilinear elliptic problem

\begin{equation*}-\Delta_{p,v} u+b(x)|\nabla u|^{p-1} \ge c(x) u^{p-1}\quad \text{in}\ \Omega,\end{equation*}

with general weights $b\ge0$ and c > 0. Liouville-type results are also discussed for a class of higher order differential equations.

In this article, we deal with non-existence results, i.e., Liouville type results, for positive radial solutions of quasilinear elliptic equations with weights both in the entire $\mathbb R^N$ and in a ball, in the latter case under Dirichlet boundary conditions. The presence of weights, possibly singular or degenerate, makes the study fairly delicate. The proofs use a Pohozaev type identity combined with an accurate qualitative analysis of solutions. In the last part of the article, a non-existence theorem is proved for a Dirichlet problem with a convection term.

We consider positive solutions to a class of quasilinear elliptic problems involving the Hardy potential under zero Dirichlet boundary condition. Via moving plane method, proving a weak comparison principle, we prove symmetry and monotonicity properties for the solutions defined on strictly convex symmetric domains.

In this paper, we obtain gradient estimates of the positive solutions to weighted p-Laplacian type equations with a gradient-dependent nonlinearity of the form0.1

$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$

$\Omega \subseteq {\open R}^N$

$N\ges 2$

$m,q\in [0,\infty )$

$1<p\les N+\sigma $

$q>\max \{p-m-1,\sigma +\tau -1\}$

$ \vert \nabla u \vert $

$ \vert \nabla u \vert $

Let 1 < p < +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type

$-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

In the present paper we deal with the existence of multiple solutions for a quasilinear elliptic problem involving an arbitrary perturbation. Our approach, based on an abstract result of Ricceri, combines truncation arguments with Moser-type iteration technique.

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for a class of nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler’s strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This contraction implies linear convergence of the discrete solutions to the exact solution in the usual H1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.