Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

Liouville type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: −aij(x)∂iju(x)− qi(x)∂iu(x) = f(x, u(x)), x ∈ R . The aim is to prove uniqueness of positive bounded solutions Liouville type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the line...

Liouville type theorems and Harnack type inequalities for semilinear elliptic equations

Palais-smale Approaches to Semilinear Elliptic Equations in Unbounded Domains

Let Ω be a domain in RN , N ≥ 1, and 2∗ = ∞ if N = 1, 2, 2∗ = 2N N−2 if N > 2, 2 < p < 2 ∗. Consider the semilinear elliptic problem −∆u+ u = |u|p−2u in Ω; u ∈ H 0 (Ω). Let H1 0 (Ω) be the Sobolev space in Ω. The existence, the nonexistence, and the multiplicity of positive solutions are affected by the geometry and the topology of the domain Ω. The existence, the nonexistence, and the multipli...

Dirichlet Problems for Semilinear Elliptic Equations with a Fast Growth Coefficient on Unbounded Domains

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations that are similar either to the singular Emden-Fowler equation or a sublinear elliptic equation. The result obtained can be applied to equations with coefficients of the nonlinear term growing exponentially. The proof is based on the super an...

Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: two approaches

In this paper, we study the asymptotic behavior as x1 → +∞ of solutions of semilinear elliptic equations in quarteror half-spaces, for which the value at x1 = 0 is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum prin...