Liouville type theorems for stable solutions of the weighted elliptic system

Liouville-type theorems for stable and finite Morse index solutions of a quasi-linear elliptic equation

We establish Liouville-type theorems for stable and finite Morse index weak solutions of −∆pu = f(x)F (u) in R . For a general non-linearity F ∈ C(R) and f(x) = |x|, we prove such theorems in dimensions N ≤ 4(p+α) p−1 +p, for bounded radial stable solutions. Then, we give some point-wise estimates for not necessarily bounded solutions. Also, similar theorems will be proved for both radial finit...

Liouville theorems for stable solutions of biharmonic problem

We prove some Liouville type results for stable solutions to the biharmonic problem ∆u = u, u > 0 in R where 1 < q < ∞. For example, for n ≥ 5, we show that there are no stable classical solution in R when n+4 n−4 < q ≤ ( n−8 n )−1 + .

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumption of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma devel...

Integral representations and Liouville theorems for solutions of periodic elliptic equations

The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arriv...

Liouville type theorems and Harnack type inequalities for semilinear elliptic equations