Positive and sign-changing solutions for a quasilinear Steklov nonlinear boundary problem with critical growth - Archive ouverte HAL Access content directly
Journal Articles Nonlinear Differential Equations and Applications Year : 2021

Positive and sign-changing solutions for a quasilinear Steklov nonlinear boundary problem with critical growth

Abstract

In this work we study the existence of positive solutions and nodal solutions for the following p-laplacian problem with Steklov boundary conditions on a bounded regular domain Ω ⊂ R N , −∆ p u + V (x)|u| p−2 u = 0 in Ω; |∇u| p−2 ∂u ∂ν = λa(x)|u| p−2 u + b(x)|u| p * −2 u on ∂Ω; with given numbers p, N satisfying 1 < p < N , p * := p(N −1) N −p the critical exponent for the Sobolev trace map W 1,p (Ω) → L q (∂Ω) and functions b 0 and a, V possibly indefinite. By minimization on subsets of the associated Nehari manifold, we prove the existence of positive solutions if N ≥ max{2p− 1, 3} and the parameter λ close to the principal eigenvalues of the operator −∆ p + V with weighted-Steklov boundary conditions. We also prove the existence on nodal solutions for a definite and N > max{p 2 , 2p, p p−1 , 2}. Our results show striking differences between the cases p > 2, p = 2 and p < 2.
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Dates and versions

hal-04121044 , version 1 (07-06-2023)

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Mabel Cuesta, Liamidi Leadi. Positive and sign-changing solutions for a quasilinear Steklov nonlinear boundary problem with critical growth. Nonlinear Differential Equations and Applications, 2021, 28 (1), pp.3. ⟨10.1007/s00030-020-00666-4⟩. ⟨hal-04121044⟩
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