Existence Results for the p-Laplacian with Nonlinear Boundary Conditions

Bonder, Julián Fernández; Rossi, Julio D.

doi:10.1006/jmaa.2001.7609

Cited by 106 publications

(45 citation statements)

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“…Regarding existence and multiplicity of solutions to (1.2) we point out (without guarantee of completeness) the papers in [20], [21], [22], [30], [32], [37], [46], [48], and the references therein. Referring to homogeneous Neumann problems, the existence of at least three solutions in case p > N was shown with different methods for example in [1], [5] and [6] (see also [7] for infinitely many solutions) while the more complicated case p ≤ N was recently studied in [19].…”

Section: Introductionmentioning

confidence: 99%

Multiplicity results to a class of variational-hemivariational inequalities

Bonanno¹,

Winkert²

2016

TMNA

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Abstract. This paper deals with variational-hemivariational inequalities involving the p-Laplace operator and a nonlinear Neumann boundary condition. Based on an abstract critical point result, which is developed at the beginning of the paper, it is shown the existence of at least three solutions to such inequalities whereby the cases p > N and p ≤ N are treated separately. The applicability of these results is emphasized with suitable examples.

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Section: Introductionmentioning

confidence: 99%

Multiplicity results to a class of variational-hemivariational inequalities

Bonanno¹,

Winkert²

2016

TMNA

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“…Without guarantee of completeness we refer to the papers of Abreu-Marcos dó O-Medeiros [1], Fernández Bonder-Rossi [6], Fernández Bonder [7], [8], Li-Li [14], Liu-Zheng [16], Martínez-Rossi [18], Winkert [23], [25], Zhao-Zhao [27], and the references therein. To the best of our knowledge, the results presented here are the first By…”

Section: Introductionmentioning

confidence: 99%

Parametric nonlinear nonhomogeneous Neumann equations involving a nonhomogeneous differential operator

2014

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Abstract. This work is concerned with the existence of solutions to parametric elliptic equations driven by a nonhomogeneous differential operator with a nonhomogeneous Neumann boundary condition. The assumptions on the operator involve the p-Laplacian, the (p, q)-Laplacian, and the generalized pmean curvature differential operator. Based on variational tools combined with truncation and comparison techniques we prove the existence of at least three nontrivial solutions provided the parameter is sufficiently large whereby the first solution is positive, the second one is negative and the third one has changing sign (nodal). IntroductionLet Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω and let 1 < q ≤ p. We study the following nonlinear nonhomogeneous Neumann problemwhere ∂u/∂n a denotes the conormal derivative with respect to the mapping a : R N → R N which is supposed to be continuous and strictly monotone with (p − 1)-growth. The nonlinearities f : Ω × R → R and h : ∂Ω × R → R are assumed to be Carathéodory functions being (p − 1)-superlinear near ±∞ and bounded on bounded sets while χ, λ are real parameters to be specified. The aim of this paper is to establish the existence of at least three nontrivial solutions provided λ > 0 is sufficiently large depending on the first two eigenvalues of the negative q-Laplacian −∆ q with Steklov boundary condition. In addition we give complete sign information of the solutions obtained, that is, the first solution is positive, the second one is negative and finally, the third one has changing sign.Such results are known for quasilinear elliptic equations involving the p-Laplacian and were obtained by a number of authors in the last years with different methods. (H) There exists a number δ f > 0 such that f (x, s) |s| p−2 s ≥ 0 for all 0 < |s| ≤ δ f and for a.a. x ∈ Ω. Assumption (H) implies that the function f must change sign near zero. Now, we do not need this condition on f . It is also worth pointing out that we do not need differentiability, polynomial growth or some integral conditions on the mappings f and h. Our approach is based on variational methods coupled with truncation and comparison techniques. Preliminaries and hypothesesIn this section we recall some basic facts about critical point theory which will be needed in Section 3. For this purpose, let X be a Banach space with norm · X and denote by X * its dual space equipped with the dual norm · X * , that iswhere ·, · (X * ,X) stands for the duality paring of (X * , X).Definition 2.1. The functional ϕ ∈ C 1 (X) fulfills the Palais-Smale condition at the level c ∈ R (the PS c -condition for short) if every sequence (u n ) n≥1 ⊆ X satisfying ϕ(u n ) → c and ϕ (u n ) → 0 in X * , admits a strongly convergent subsequence. We say that ϕ satisfies the Palais-Smale condition (the PS-condition for short) if it satisfies the PS c -condition for every c ∈ R.This compactness-type condition on ϕ leads to a deformation theorem which is the main ingredient in the minimax theory of the critical values of ϕ. A basic r...

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“…In recent years, the existence of solutions for the semilinear/quasilinear elliptic equations with nonlinear boundary conditions have been widely studied (see, e.g., [9,12,16,17,28,36,37,41] and the references therein). In particular, in [37], the authors studied the multiple solutions of the following systems:…”

Section: Introductionmentioning

confidence: 99%