Existence and nonexistence of radial positive solutions of superlinear elliptic systems

The main goal in this paper is to prove the existence of radial positive solutions of the quasilinear elliptic system [formula], where [omega] is a ball in RN and f, g are positive continuous functions satisfying f(x, 0, 0) = g(x, 0, 0) = 0 and some growth conditions which correspond, roughly speaking, to superlinear problems. Two different sets of conditions, called strongly and weakly coupled, are given in order to obtain existence. We use the topological degree theory combined with the blow up method of Gidas and Spruck. When [omega] = RN, we give some sufficient conditions of nonexistence of radial positive solutions for Liouville systems

Nonexistence results for nonlocal equations with critical and supercritical nonlinearities

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form $$Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.$$ These operators are infinitesimal generators of symmetric L\'evy processes. Our results apply to even kernels $K$ satisfying that $K(y)|y|^{n+\sigma}$ is nondecreasing along rays from the origin, for some $\sigma\in(0,2)$ in case $a_{ij}\equiv0$ and for $\sigma=2$ in case that $(a_{ij})$ is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for $L$ in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to $n$ and $\sigma$). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian $(-\Delta)^s$ (here $s>1$) or the fractional $p$-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin

Existence and blow-up of global solutions for a class of fractional Lane-Emden heat flow system

In this paper, we consider a class of Lane–Emden heat flow system with the fractional Laplacian ut + (−∆) 2 u = N1(v) + f1(x), (x, t) ∈ Q, vt + (−∆) 2 v = N2(u) + f2(x), (x, t) ∈ Q, u(x, 0) = a(x), v(x, 0) = b(x), x ∈ RN, where 0 < α ≤ 2, N ≥ 3, Q := RN × (0, +∞), fi(x) ∈ L 1 loc(RN) (i = 1, 2) are nonnegative functions. We study the relationship between the existence, blow-up of the global solutions for the above system and the indexes p, q in the nonlinear terms N1(v), N2(u). Here, we first establish the existence and uniqueness of the global solutions in the supercritical case by using Duhamel’s integral equivalent system and the contraction mapping principle, and we further obtain some relevant properties of the global solutions. Next, in the critical case, we prove the blow-up of nonnegative solutions for the system by utilizing some heat kernel estimates and combining with proof by contradiction. Finally, by means of the test function method, we investigate the blow-up of negative solutions for the Cauchy problem of a more general higher-order nonlinear evolution system with the fractional Laplacian in the subcritical case

Existence and uniqueness of positive large solutions to some cooperative elliptic systems

In this work we consider positive solutions to cooperative elliptic systems of the form −∆u = λu−u2 +buv, −∆v = µv −v2 +cuv in a bounded smooth domain Ω ⊂ RN (λ, µ ∈ R, b, c > 0) which blow up on the boundary ∂Ω, that is u(x), v(x) → +∞ as dist(x, ∂Ω) → 0. We show existence and nonexistence of solutions, and give sufficient conditions for uniqueness. We also provide an exact estimate of the behaviour of the solutions near the boundary in terms of dist(x, ∂Ω).Ministerio de Ciencia y Tecnologí