Abstract
We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz‐Sobolev space.
Highlights
Let X be a separable and reflexive real Banach space; let I : X → R be a coercive, sequentially weakly lower semicontinuous C1 functional, belonging to WX, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X∗; J : X → R a C1 functional with compact derivative
Let X be a reflexive real Banach space; S ⊂ R an interval, let I : X → R be a sequentially weakly lower semicontinuous C1 functional, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X∗; J : X → R a C1 functional with compact derivative
For each compact interval a, b ⊂ γ, ∞, there exists B > 0 with the following property: for every λ ∈ a, b and g, there exists δ > 0 such that, for each μ ∈ 0, δ, the problem Pλ,μ has at least three weak solutions whose norms in X are less than B
Summary
Consider the following nonhomogeneous Neumann problem with a perturbed term:. 0, on ∂Ω, Pλ,μ where Ω is a bounded domain in RN N ≥ 3 with smooth boundary ∂Ω, ν is the outer normal to ∂Ω, f, g : Ω × R → R are two Caratheodory functions, λ > 0, μ ≥ 0 are two parameters, and the function α : 0, ∞ → R is such that φ t : R → R defined by φt α |t| t, t / 0, 0, t 0, is an odd, strictly increasing homeomorphism from R to R. Consider the following nonhomogeneous Neumann problem with a perturbed term:. Since the operator in the divergence form is nonhomogeneous, we introduce Orlicz-Sobolev space which is an appropriate setting for these problems. Such space originated with Nakano 5 and was developed by Musielak and Orlicz 6. To the best of our knowledge, for the perturbed nonhomogeneous Neumann problem, there has so far been few papers concerning its multiple solutions. Motivated by the above facts, in this paper, we establish some new sufficient conditions under which such a problem possesses three weak solutions in Orlicz-Sobolev space.
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