On some quasilinear equations involving the p-Laplacian with Robin boundary conditions

Abstract

Let Ω ⊂ ℝ ^N be a smooth-bounded domain, let f and g be continuous functions on , g ⩾ 0. Let a > 0 be some real number and let p ∈ (1, N ) and q ∈ (p, p*] be two real numbers, where p* = np/(n − p). We are interested in proving the results of existence and non-existence for the non-negative solution of the equation

We consider the case λ < λ₁ (coercive case) and the case λ ⩾ λ₁, where λ₁ is the principal eigenvalue for the operator Δ _p  + g with Robin boundary conditions from which we recall the definition and some properties.

Keywords:

• p-Laplacian
• Robin boundary conditions
• critical Sobolev exponent

AMS Subject Classifications:

• 35J60
• 35D30

Acknowledgements

I would like to thank Françoise Demengel for introducing this subject and for her helpful remarks on my work.