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Bifurcation theorems for nonlinear problems with lack of compactness

Francesca Faraci, Roberto Livrea (2003)

Annales Polonici Mathematici

We deal with a bifurcation result for the Dirichlet problem ⎧ - Δ p u = μ / | x | p | u | p - 2 u + λ f ( x , u ) a.e. in Ω, ⎨ ⎩ u | Ω = 0 . Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number λ * μ such that for every λ ] 0 , λ * μ [ the above problem admits a nonzero weak solution u λ in W 1 , p ( Ω ) satisfying l i m λ 0 | | u λ | | = 0 .

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system Δ p u = g ( u - α v ) , Δ p v = f ( v - β u ) in a smooth bounded domain of N , where Δ p is the p-Laplacian operator defined by Δ p u = d i v ( | u | p - 2 u ) with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

Dimension reduction for −Δ1

Maria Emilia Amendola, Giuliano Gargiulo, Elvira Zappale (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A 3D-2D dimension reduction for −Δ1 is obtained. A power law approximation from −Δp as p → 1 in terms of Γ-convergence, duality and asymptotics for least gradient functions has also been provided.

Entropy solutions for nonhomogeneous anisotropic Δ p ( · ) problems

Elhoussine Azroul, Abdelkrim Barbara, Mohamed Badr Benboubker, Hassane Hjiaj (2014)

Applicationes Mathematicae

We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet p -Laplacian and the Navier p -biharmonic operator on a ball of radius R in N and its asymptotics for p approaching 1 and . Let p tend to . There is a critical radius R C of the ball such that the principal eigenvalue goes to for 0 < R R C and to 0 for R > R C . The critical radius is R C = 1 for any N for the p -Laplacian and R C = 2 N in the case of the p -biharmonic operator. When p approaches 1 , the principal eigenvalue of the Dirichlet...

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ - d i v ( | x | - a p | u | p - 2 u ) + h ( x ) | u | p - 2 u = g ( x ) | u | r - 2 u , x ∈ Ω, ⎨ ⎩ | x | - a p | u | p - 2 u / ν = λ f ( x ) | u | q - 2 u , x ∈ ∂Ω, where Ω is an exterior domain in N , that is, Ω = N D , where D is a bounded domain in N with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method,...

Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces

Nguyen Thanh Chung (2015)

Annales Polonici Mathematici

We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where Ω N , N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, ρ ( u ) = Ω ( Φ ( | u | ) + Φ ( | u | ) ) d x , M: [0,∞) → ℝ is a continuous function, K L ( Ω ) , and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

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