Nonexistence of positive supersolutions of elliptic equations via the maximum principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.Comment: revised version, 32 page

Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

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 domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$ for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast with exterior domains depends both on the geometry of the domain $G$ and the coefficients $a$ of the equation.Comment: 20 page

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\Delta_p u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $\mu\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,\sigma)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Pr\"ufer-Transformation.Comment: 34 pages, 1 figur

Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems

We study qualitative properties of positive solutions of noncooperative, possibly nonvariational, elliptic systems. We obtain new classification and Liouville type theorems in the whole Euclidean space, as well as in half-spaces, and deduce a priori estimates and existence of positive solutions for related Dirichlet problems. We significantly improve the known results for a large class of systems involving a balance between repulsive and attractive terms. This class contains systems arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions.Comment: 35 pages, to appear in Archive Rational Mech. Ana

Nonexistence of stable solutions to quasilinear elliptic equations on Riemannian manifolds

We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls

Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

We study fully nonlinear elliptic equations such as $$F(D^2u) = u^p, \quad p>1,$$ in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of $F$, that sharply characterizes the range of $p>1$ for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-V\'eron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space $\R^n$, in case $-F$ is Laplace's operator and Pucci's operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.Comment: 16 pages, new existence results adde

Symmetries, Hopf fibrations and supercritical elliptic problems

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 exponents $p>\frac{2N}{N-2}.$ Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents

Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.Comment: Published online in Calculus of Variations and Partial Differential Equation