On Neumann Boundary Value Problems For Some Quasilinear Elliptic Equations (original) (raw)

Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R N

Milan Journal of Mathematics, 2014

We establish a result on the existence of a positive solution for the following class of degenerate quasilinear elliptic problems: (P) −Δ ap u + V (x)|x| −ap * |u| p−2 u = K(x)f (x, u), in R N , u > 0, in R N , u ∈ D 1,p a (R N), where −Δ ap u = −div(|x| −ap |∇u| p−2 ∇u), 1 < p < N, −∞ < a < N −p p , a ≤ e ≤ a + 1, d = 1 + a − e, and p * := p * (a, e) = Np N −dp denotes the Hardy-Sobolev's critical exponent, V and K are bounded nonnegative continuous potentials, K vanishes at infinity, and f has a subcritical growth at infinity. The technique used here is the variational approach.

Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

Advanced Nonlinear Studies, 2014

Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k1ξp ≤ f (ξ) ≤ k2ξp for all ξ ≥ 0 and some k1, k2 > 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

On the existence of signed solutions for a quasilinear elliptic problem in RN

Matemática Contemporânea, 2007

We study the existence of positive and negative weak solutions for the equation −∆ p u + V (x)|u| p−2 u = λf (u) in R N , where −∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian operator, 1 < p < N , λ is a positive real parameter and the potential V : R N → R is bounded from below for a positive constant and "large" at infinity. It is assumed that the nonlinearity f : R → R is continuous and just superlinear in a neighborhood of the origin.

On positive solutions for a class of singular quasilinear elliptic systems

Journal of Mathematical Analysis and Applications, 2007

We study through the lower and upper-solution method, the existence of positive weak solution to the quasilinear elliptic system with weights ⎧ ⎪ ⎨ ⎪ ⎩ −div(|x| −ap |∇u| p−2 ∇u) = λ|x| −(a+1)p+c 1 u α v γ in Ω, −div(|x| −bq |∇v| q−2 ∇v) = λ|x| −(b+1)q+c 2 u δ v β in Ω, u = v = 0 o n ∂Ω, where Ω is a bounded smooth domain of R N , with 0 ∈ Ω, 1 < p, q < N, 0 a < N −p p , 0 b < N −q q , 0 α < p − 1, 0 β < q − 1, δ, γ, c 1 , c 2 > 0 and θ := (p − 1 − α)(q − 1 − β) − γ δ > 0, for each λ > 0.

Existence of nonnegative solutions to singular elliptic problems, a variational approach

Discrete & Continuous Dynamical Systems - A, 2018

We consider the problem −∆u = χ {u>0} g (., u) + f (., u) in Ω, u = 0 on ∂Ω, u ≥ 0 in Ω, where Ω is a bounded domain in R n , f : Ω×[0, ∞) → R and g : Ω × (0, ∞) → [0, ∞) are Carathéodory functions, with g (x, .) nonnegative, nonincreasing, and singular at the origin. We establish sufficient conditions for the existence of a nonnegative weak solution 0 ≡ u ∈ H 1 0 (Ω) to the stated problem. We also provide conditions that guarantee that the found solution is positive a.e. in Ω. The problem with a parameter ∆u = χ {u>0} g (., u)+λf (., u) in Ω, u = 0 on ∂Ω, u ≥ 0 in Ω is also studied. For both problems, the special case when g (x, s) := a (x) s −α(x) , i.e., a singularity with variable exponent, is also considered.