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Multiple solutions to a perturbed Neumann problem

100%

Cordaro G.

Studia Mathematica

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2007

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tom 178

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nr 2

167-175

EN

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 $ℝ^{N}$ with boundary of class C², $α ∈ L^{∞}(Ω)$ with $ess inf_{Ω}α > 0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $sup_{|s|≤t} |g(⋅,s)| ∈ L^{p}(Ω)$ and $g(⋅,t) ∈ L^{∞}(Ω)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2 ξ² - ∫_{0}^{ξ} f(t)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above Neumann problem has at least M+integer part of M/2 distinct strong solutions in $W^{2,p}(Ω)$.

2

An existence and localization theorem for the solutions of a Dirichlet problem

63%

Anello G., Cordaro G.

Annales Polonici Mathematici

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2004

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tom 83

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nr 2

107-112

EN

We establish an existence theorem for a Dirichlet problem with homogeneous boundary conditions by using a general variational principle of Ricceri.

3

Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

63%

Anello G., Cordaro G.

Colloquium Mathematicum

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2003

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tom 97

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nr 2

221-231

EN

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-Δ_{p}u + λ(x)|u|^{p-2}u = μf(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $Ω ⊂ ℝ^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $λ ∈ L^{∞}(Ω)$ with $essinf_{x∈Ω} λ(x) > 0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Colloquium Mathematicum

1 Studia Mathematica

3 Cordaro G.

2 Anello G.

1 2007

1 2004

1 2003

Multiple solutions to a perturbed Neumann problem

An existence and localization theorem for the solutions of a Dirichlet problem

Infinitely many positive solutions for the Neumann problem involving the p-Laplacian