Study to the Resonance of $ p $-Laplacian problem with mixed boundary

2011

In the first part of the article, we give necessary and sufficient conditions for the solvability of a class of nonlinear elliptic boundary value problems with nonlinear boundary conditions involving the q-Laplace-Beltrami operator. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L ∞ -setting.

Nodea-nonlinear Differential Equations and Applications, 2008

In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole $\mathbb {R}^N$ . Via variational arguments, we obtain existence and regularity of nontrivial solutions.

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2015

This article shows the existence of weak solutions of a resonance problem for uniformly p-Laplacian system in a bounded domain in R N. Our arguments are based on the Saddle Point Theorem (P.H.Rabinowitz) and rely on a generalization of the Landesman-Lazer type condition.

Boletim da Sociedade Paranaense de Matemática, 2011

We consider elliptic equations of p-Laplacian type with the nonlinear boundary condition of the form −∆pu + |u| p−2 u = λf 1 (u) + µg 1 (u) in Ω, |∇u| p−2 ∂u ∂n = λf 2 (u) + µg 2 (u) in ∂Ω, where Ω ⊂ R N (N ≧ 3) is a bounded domain with smooth boundary ∂Ω, ∂ ∂n is the outer unit normal derivative, λ, µ are parameters. The functions f i , i = 1, 2, are assumed to be (p − 1)-sublinear while g i , i = 1, 2, are (p − 1)-assymptotically linear at infinity. Using variational techniques, an existence result is given.

Boletim da Sociedade Paranaense de Matemática, 2016

In this paper we study the existence of at least two nontrivial solutions for the nonlinear problem p-Laplacian, with nonlinear boundary conditions. We establish that there exist at least two solutions, which are opposite signs. For this reason, we characterize the first eigenvalue of an intermediary eigenvalue problem by the minimization method. In fact, in some sense, we establish the non-resonance below the first eigenvalues of nonlinear Steklov-Robin.

2016

This work concerns the multi-point nonlinear Neumann boundaryvalue problem involving a p-Laplacian-like operator (φ(u′))′ = f(t, u, u′), t ∈ (0, 1), u′(0) = u′(η), φ(u′(1)) = m X

Nonlinear Differential Equations and Applications NoDEA, 2008

In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole R N . Via variational arguments, we obtain existence and regularity of nontrivial solutions.

Journal of Mathematical Analysis and Applications, 2004

This paper deals with two mixed nonlinear boundary value problems depending on a parameter λ. For each of them we prove the existence of at least three generalized solutions when λ lies in an exactly determined open interval. Usefulness of this information on the interval is then emphasized by means of some consequences.

Nonlinear Analysis: Theory, Methods & Applications, 2004

In this paper we prove that there exists a first curve of the Fučik spectrum of the problem ∆ p u = |u| p-2 u in Ω with a nonlinear boundary condition given by |∇u| p-2 ∂u/∂ν = α(u + ) p-1 -β(u -) p-1 on the boundary of the domain. We also prove that there exists a sequence of curves of the Fučik spectrum which exist locally in the neighborhood of suitable eigenvalues of the p-laplacian with a nonlinear boundary condition. Finally, we study a resonance problem with respect to the Fučik spectrum. This paper is devoted to find existence results for elliptic problems involving the p-laplacian with a nonlinear flux boundary condition. More precisely, we look for solutions of reso.1

Journal of Mathematical Analysis and Applications, 2001

In this paper we study the existence of nontrivial solutions for the problem, ∆pu = |u| p−2 u in a bounded smooth domain Ω ⊂ R N , with a nonlinear boundary condition given by |∇u| p−2 ∂u/∂ν = f (u) on the boundary of the domain. The proofs are based on variational and topological arguments.