Related papers: On the H\'enon-Lane-Emden conjecture

In the present paper we derive Liouville type results and existence of periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates…

In this paper, we study the following Lane-Emden system with nearly critical non-power nonlinearity \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{lll} -\Delta u =\frac{|v|^{p-1}v}{[\ln(e+|v|)]^\epsilon}\ \ &{\rm in}\ \Omega,…

We prove Liouville theorem for the equation $\Delta v + N v^p + M |\nabla v|^{q}= 0$ in $\mathbb R^n$, with $M, N > 0, q = \frac{2p}{p + 1}$ in the critical and subcritical case. The proof is based on a differential identity and Young…

We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic…

We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation $ -\Delta u= f(u)$ in the upper half-space $\mathbb{R}^{N}_{+}$. Some Liouville-type theorems are also proven in the…

We prove a Liouville type theorem for arbitrarily growing positive viscosity supersolutions of fully nonlinear uniformly elliptic equations in halfspaces. Precisely, let $M^-_{\lambda, \Lambda}$ be the Pucci's inf- operator, defined as the…

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in $\mathbb{R}^{n}$ with sign-changing nonlinearity $f$. Under the hypothesis that the equation does not have any…

In this paper, we consider the critical Lane-Emden system \begin{align*} \begin{cases} -\Delta u=K_1(y)v^p,\quad y\in \mathbb{R}^N,&\\ -\Delta v=K_2(y)u^q,\quad y\in \mathbb{R}^N,&\\ u,v>0, \end{cases} \end{align*} where $N\geq 5$, $p,q\in…

We examine the system given by \hfill -\Delta u = \lambda (v+1)^p \qquad \Omega \hfill -\Delta v = \gamma (u+1)^\theta \qquad \Omega, \hfill u = v =0 \qquad \quad \partial \Omega, where $ \lambda,\gamma$ are positive parameters and where $…

We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $\Omega\subset \mathbb{R}^N$. In particular, we provide a sharp decay criterion that…

We study the Liouville type problem for the stationary 3D Navier-Stokes equations on $\Bbb R^3$. Specifically, we prove that if $v$ is a smooth solution to (NS) satisfying $\omega={\rm curl}\,v \in L^q (\Bbb R^3) $ for some $\frac32 \leq q<…

We prove that if the elliptic problem $-\Delta u+b(x)|\nabla u|=c(x)u$ with $c\ge0$ has a positive supersolution in a domain $\Omega$ of $ \IR^{N\ge 3}$, then $c,b$ must satisfy the inequality \[\sqrt{ \int_\Omega c\phi^2}\le \sqrt{…

In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u\in H^{1,p}_0(\Omega) $$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has…

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like…

Let $n$ be an integer and $s$ be a real number such that $n > 2s \geq 2$. Inspired by the perturbation approach initiated by F. Hang and P. Yang (\textit{Int. Math. Res. Not. IMRN}, 2020), we are interested in non-negative, smooth solution…

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x)…

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have…

We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ in 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…