Positive and sign-changing solutions for a quasilinear Steklov nonlinear boundary problem with critical growth
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Nonlinear Differential Equations and Applications NoDEA
Positive and sign-changing solutions for a quasilinear Steklov nonlinear boundary problem with critical growth Mabel Cuesta and Liamidi Leadi 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 ⊂ RN , −Δp u + V (x)|u|p−2 u = 0 in Ω; |∇u|p−2 ∂u = λa(x)|u|p−2 u + b(x)|u|p∗ −2 u on ∂Ω; ∂ν −1) with given numbers p, N satisfying 1 < p < N . Here p∗ := p(N is N −p 1,p q the critical exponent for the Sobolev trace map W (Ω) → L (∂Ω), the functions b 0 and a, V are 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 −Δp +V with weighted-Steklov boundary conditions. We also prove the existence on nodal solutions for a definite and p , 2}. Our results show striking differences between N > max{p2 , 2p, p−1 the cases p > 2, p = 2 and p < 2.
Mathematics Subject Classification. 35D05, 35J60, 35J65, 35J70, 35J25, 35J35. Keywords. Critical growth, Indefinite weights, Steklov boundary conditions, p-Laplacian operator.
1. Introduction Consider the following problem of parameter λ in Ω; −Δp u + V (x)|u|p−2 u = 0 p−2 p∗ −2 |∇u|p−2 ∂u = λa(x)|u| u + b(x)|u| u on ∂Ω; ∂ν
(1.1)
for 1 < p < N , a, b two given functions in C γ (∂Ω) for some γ > 0, a ≡ 0 with −1) N b ≥ 0, V ∈ L∞ (Ω) and p∗ := p(N N −p . The domain Ω is a bounded subset of R 0123456789().: V,-vol
3
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M. Cuesta and L. Leadi
NoDEA
of class C 2,α for some 0 < α < 1 and N ≥ 3. Our aim is to prove the existence of solutions for λ close to the principal eigenvalues of (1.5) (see below). In the case a ≡ 0, V ≡ 1, b ≡ 1, the quasilinear problem (1.1) arises, for instance, when searching for functions u ∈ W 1,p (Ω) for which the norm of the Sobolev’s trace immersion ip∗ ,Ω : W 1,p (Ω) → Lp∗ (∂Ω) is achieved: (|∇u|p + |u|p ) dx Ω −p inf 1,p S0 := ip∗ ,Ω = (1.2) p/p∗ , u∈W 1,p (Ω)\W0 (Ω) |u|p∗ dσ ∂Ω where σ is the restriction to ∂Ω of the the (N − 1)-Hausdorff measure, which coincides with the usual Lebesgue surface measure as ∂Ω is regular enough. Due to the lack of compactness of ip∗,Ω , the existence of minimizers for (1.2) does not follows by standards methods. Following the ideas of [2,8] and [5], Fernandez-Bonder and Rossi proved in [7] that a sufficient condition for the −1 where existence of minimizers for (1.2) is that S0 < KN,p −1 def p p N p∗ |∇u| dx; |∇u| ∈ L (R+ ) and |u| dy = 1 . (1.3) KN,p = inf RN +
RN −1
In the linear case, i.e. p = 2, with b ≡ 1 and V ≡ 0, namely, for the problem ⎧ in Ω ⎨ Δu = 0 u>0 in Ω (Y ) ⎩ ∂u N −2 2∗ −1 on ∂Ω, ∂ν + 2 βu = u which is related to the Yamab´e problem when β = cte = mean curvature of ∂Ω, Adimurthi–Yadava [1] proved that problem (Y ) has solution when β ∈ C 1 (∂Ω), N ≥ 3 and there exists a point x0 ∈ ∂Ω such that β(x0 ) < h(x0 ) :=
N −1 1 νi ,
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