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Multiple solutions to a perturbed Neumann problem

Giuseppe Cordaro (2007)

Studia Mathematica

We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in N with boundary of class C², α L ( Ω ) with e s s i n f Ω α > 0 , f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, s u p | s | t | g ( , s ) | L p ( Ω ) and g ( , t ) L ( Ω ) for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function 1 / 2 ξ ² - 0 ξ f ( t ) d t has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above...

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