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

Abstract

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients −∇ · a · ∇u = u p ( * ), p > 1, in an unbounded cone-like domain G ⊂ R N (N ≥ 3). We prove that the critical exponent p * (a, G) = inf{p > 1 : ( * ) has a positive supersolution at infinity in G } for a nontrivial conelike 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.

Positive super-solutions to semi-linear second-order non-divergence type elliptic equations in exterior domains

“…In particular, in [13] it was shown that the critical exponent p * = N N −2 is stable with respect to the change of the Laplacian by a second-order uniformly elliptic divergence type operator with measurable coefficients, perturbed by a potential, for a sufficiently wide class of potentials (see also [14] for equations of type (1.3) in exterior domains in the presence of first order terms). In [15,16] equation (1.3) and the corresponding equation with the divergence type elliptic operators was studied on cone-like domains, and it was shown that the value of the critical exponent is dependent on the geometry of the domain even in the case of the Laplacian, and on the behaviour of the coefficients of the elliptic operator at infinity. The method developed in these papers shows that the exact value of the critical exponent can be determined based upon the precise asymptotics at infinity of the minimal harmonic function with respect to the operator (Green's function) and the "maximal" harmonic function.…”

Best possible estimates of weak solutions of boundary value problems for quasi-linear elliptic equations in unbounded domains

“…The problem of the existence and nonexistence of positive solutions to a weak linear second-order divergence type elliptic equation in an unbounded cone-like domains was studied in [7].…”

“…A number of mathematicians [2,5,7,13,15,16] took the study of quasilinear elliptic problems in unbounded domains. The problem of the existence and nonexistence of positive solutions to a weak linear second-order divergence type elliptic equation in an unbounded cone-like domains was studied in [7].…”

Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

“…The problem of existence and nonexistence of positive solutions to a weak linear second-order divergence type elliptic equation in an unbounded cone-like domain was studied in [6].…”

“…After creating the linear theory, a number of mathematicians [4,6,[8][9][10] took the study of semi-linear elliptic problems in unbounded domains. The problem of existence and nonexistence of positive solutions to a weak linear second-order divergence type elliptic equation in an unbounded cone-like domain was studied in [6].…”