The Fredholm Alternative for the p-Laplacian: Bifurcation from Infinity, Existence and Multiplicity (original) (raw)

BIFURCATION FROM THE FIRST EIGENVALUE OF THE p-LAPLACIAN WITH NONLINEAR BOUNDARY CONDITION

2019

We consider the problem ∆pu = |u|p−2u in Ω, |∇u|p−2 ∂u ∂ν = λ|u|p−2u+ g(λ, x, u) on ∂Ω, where Ω is a bounded domain of RN with smooth boundary, N ≥ 2, and ∆p denotes the p-Laplacian operator. We give sufficient conditions for the existence of continua of solutions bifurcating from both zero and infinity at the principal eigenvalue of p-Laplacian with nonlinear boundary conditions. We also prove that those continua split on two, one containing strictly positive and the other containing strictly negative solutions. As an application we deduce results on anti-maximum and maximum principles for the p-Laplacian operator with nonlinear boundary conditions.

Nonlinear Eigenvalue Problems and Bifurcation for Quasi-Linear Elliptic Operators

Mediterranean Journal of Mathematics, 2022

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to L^{\infty }L∞,whichimpliesL ∞ , which impliesL∞,whichimpliesC^{1,\alpha }$$ C 1 , α smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.

Multiple Solutions for p-Laplacian Type Problems with an Asymptotically p-linear Term

Analysis and Topology in Nonlinear Differential Equations, 2014

The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem    − div(A(x, u) |∇u| p−2 ∇u) + 1 p At(x, u) |∇u| p = f (x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R N is a bounded domain, N ≥ 2, p > 1, A is a given function which admits partial derivative At(x, t) = ∂A ∂t (x, t) and f is asymptotically p-linear at infinity. Under suitable hypotheses both at the origin and at infinity, and if A(x, •) is even while f (x, •) is odd, by using variational tools, a cohomological index theory and a related pseudo-index argument, we prove a multiplicity result if p > N in the non-resonant case.

p-Laplacian problems with nonlinearities interacting with the spectrum

Nonlinear Differential Equations and Applications NoDEA, 2013

The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problem −Δpu = g(x, u) in Ω, u = 0 on ∂Ω, where 1 < p < +∞, Δpu = div(|∇u| p−2 ∇u), Ω is an open bounded domain of R N (N ≥ 3) with smooth boundary ∂Ω and the nonlinearity g behaves as u p−1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981-1012, 1983), but extended to Banach spaces, and two sequences of quasi-eigenvalues for the p-Laplacian operator as in Candela and Palmieri (Calc.

Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian

The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case relate to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem −Δpu = λ|u| p−2 u + h x, u(x); λ in Ω ; u = 0 on ∂Ω. (P) Here, Ω is a bounded domain in R N (N ≥ 1), Δpu def = div(|∇u| p−2 ∇u) denotes the Dirichlet p-Laplacian on W 1,p 0 (Ω), 1 < p < ∞, and λ ∈ R is a spectral parameter. Let μ1 denote the first (smallest) eigenvalue of −Δp. Under some natural hypotheses on the perturbation function h : Ω× R × R → R, we show that the trivial solution (0, μ1) ∈ E = W 1,p 0 (Ω) × R is a bifurcation point for problem (P) and, moreover, there are two distinct continua, Z + μ 1 and Z − μ 1 , consisting of nontrivial solutions (u, λ) ∈ E to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ1). The continua Z + μ 1 and Z − μ 1 are either both unbounded in E, or else their intersection Z + μ 1 ∩ Z − μ 1 contains also a point other than (0, μ1). For the semilinear problem (P) (i.e., for p = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union Z + μ 1 ∩ Z − μ 1 looks like (for p > 2) in an interesting particular case. Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer's original work.