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Multiple solutions to a perturbed Neumann problem

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 ${ℝ}^{N}$ with boundary of class C², $\alpha \in {L}^{\infty }\left(\Omega \right)$ with $essin{f}_{\Omega }\alpha >0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $su{p}_{|s|\le t}|g\left(\cdot ,s\right)|\in {L}^{p}\left(\Omega \right)$ and $g\left(\cdot ,t\right)\in {L}^{\infty }\left(\Omega \right)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2\xi ²-{\int }_{0}^{\xi }f\left(t\right)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough, the above...

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

Colloquium Mathematicae

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-{\Delta }_{p}u+\lambda \left(x\right){|u|}^{p-2}u=\mu f\left(x,u\right)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $\Omega \subset {ℝ}^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $\lambda \in {L}^{\infty }\left(\Omega \right)$ with $essin{f}_{x\in \Omega }\lambda \left(x\right)>0$ 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.

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

Annales Polonici Mathematici

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

On a minimax problem of Ricceri.

Journal of Inequalities and Applications [electronic only]

Further results related to a minimax problem of Ricceri.

Journal of Inequalities and Applications [electronic only]

Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems.

Abstract and Applied Analysis

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