Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of positive solutions for nonlinear singular boundary value problem with p‐Laplacian

In this paper, we study the existence of positive solutions for the following Sturm–Liouville‐like four‐point singular boundary value problem (BVP) with p‐Laplacian where ?_p(s)=|s|^p?2 s, p>1, f is a lower semi‐continuous function. Using the fixed‐point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm–Liouville‐like singular BVP with p‐Laplacian are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Q‐integral unicyclic,bicyclic and tricyclic graphs

A graph is called Q‐integral if its signless Laplacian spectrum consists of integers. In this paper, we characterize a class of k‐cyclic graphs whose second smallest signless Laplacian eigenvalue is less than one. Using this result we determine all the Q‐integral unicyclic, bicyclic and tricyclic graphs.     

A critical Kirchhoff‐type problem involving the ‐Laplacian

In this paper we provide an existence result for a nonlocal problem of Kirchhoff‐type which involves both the p‐ and the q‐Laplacian and contains a critical term. Our approach is variational: we derive the existence of one non‐trivial solution via the multidimensional mountain pass theorem.     

Mixed Boundary Value Problems for Nonlinear Elliptic Systems with p‐Structure in Polyhedral Domains

The nonlinear elliptic system is investigated on a non‐smooth domain. Mixed boundary value conditions are given. The left‐hand side of the system has p‐structure (e.g., it is the p‐Laplacian and 1 < p < ∞). Global regularity results of u and |∇u|^p/2 in fractional order Sobolev spaces are proven.     

Existence of positive solution for impulsive boundary value problem with p‐Laplacian in Banach spaces

This paper deals with the existence of positive solution for impulsive boundary value problem with p‐Laplacian in Banach spaces. There is no literature researching on p‐Laplacian boundary value problem in Banach spaces. The main difficulty that appears when passing from p = 2 to p ≠ 2 is that for p ≠ 2, it is impossible for us to find a Green's function in the equivalent integral operator because the differential operator (?_p(u ′ )) ′ is nonlinear, so it is difficult for us to prove that the equivalent integral operator is a strict‐set‐contraction operator. Even in the absence of pulse effect, these results are new. Copyright © 2012 John Wiley & Sons, Ltd.     

Certain classes of potentials for p ‐Laplacian to be non‐degenerate

Given a positive integer n and an exponent 1 ≤ α ≤ ∞. We will find explicitly the optimal bound r_n such that if the L^α norm of a potential q (t ) satisfies ‖q ‖equation/tex2gif-inf-2.gif < r_n then the n ^th Dirichlet eigenvalue of the onedimensional p ‐Laplacian with the potential q (t ): (|u ′|^{p –2} u ′)′ + (λ + q (t )) |u |^{p –2}u = 0 (1 < p < ∞) will be positive. Using these bounds, we will construct, for the Dirichlet, the Neumann, the periodic or the antiperiodic boundary conditions, certain classes of potentials q (t ) so that the p ‐Laplacian with the potential q (t ) is non‐degenerate, which means that the above equation with λ = 0 has only the trivial solution verifying the corresponding boundary condition. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive Solutions of Nonlinear Systems with <Emphasis Type="Italic">p</Emphasis>-Laplacian on Finite and Semi-infinite Intervals

Existence and localization results are established for systems of second-order differential equations with p -Laplacian on finite and semi-infinite intervals. For a compact interval, the compression and expansion conditions that are used are related to the first eigenvalue of the p-Laplacian.     

Multiple and sign‐changing solutions for the multivalued p ‐Laplacian equation

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient. Under conditions given in terms of the first eigenvalue as well as the Fu?ik spectrum of the p ‐Laplacian we prove the existence of a positive, a negative and a sign‐changing solution. Our approach is based on variational methods for nonsmooth functionals (nonsmooth critical point theory, second deformation lemma), and comparison principles for multivalued elliptic problems. In particular, the existence of extremal constant‐sign solutions plays a key role in the proof of sign‐changing solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple solutions for a Robin‐type differential inclusion problem involving the p(x)‐Laplacian

In this paper, we obtain the existence of at least two nontrivial solutions for a Robin‐type differential inclusion problem involving p(x)‐Laplacian type operator and nonsmooth potentials. Our approach is variational, and it is based on the nonsmooth critical point theory for locally Lipschitz functions. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

On the Ambrosetti‐Prodi problem for a system involving p‐Laplacian operator

In this work we study an Ambrosetti‐Prodi type problem for an elliptic system involving p‐Laplacian operator. The sub and supersolution method and the Leray‐Schauder Degree Theory are used in order to prove our result.     

Extremal solutions for p‐Laplacian boundary value problems with the right‐handed Riemann‐Liouville fractional derivative

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results.     

The existence of a nontrivial solution to p‐Laplacian equations in with supercritical growth

In this paper, we consider the p‐Laplacian equations in with supercritical growth where △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N is the p‐Laplacian operator. Under certain assumptions on V (x) and f(u) that will be given in Section 1, we prove that the problem has at least a nontrivial solution by using variational methods combined with perturbation arguments. The solutions to subcritical p‐Laplacian equations are estimated applying the L ^∞ norm. Copyright © 2012 John Wiley & Sons, Ltd.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations: II

In this paper (which is a continuation of Part‐I), we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations in the case of 1<p?2?1/N when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2001 John Wiley & Sons, Ltd.     

Multiple solutions for strongly resonant Robin problems

We consider nonlinear (driven by the p‐Laplacian) and semilinear Robin problems with indefinite potential and strong resonance with respect to the principal eigenvalue. Using variational methods and critical groups, we prove four multiplicity theorems producing up to four nontrivial smooth solutions.     

On a class of quasilinear eigenvalue problems in RN

We study an eigenvalue problem in R ^N which involves the p ‐Laplacian (p > N ≥ 2) and the nonlinear term has a global (p – 1)‐sublinear growth. The existence of certain open intervals of eigenvalues is guaranteed for which the eigenvalue problem has two nonzero, radially symmetric solutions. Some stability properties of solutions with respect to the eigenvalues are also obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Critical extinction and blow‐up exponents for fast diffusive p‐Laplacian with sources

We discuss and determine the critical extinction and blow‐up exponents for the homogeneous Dirichlet boundary value problem of the fast diffusive p‐Laplacian with sources. Copyright © 2007 John Wiley & Sons, Ltd.     

Boundary Problems for Harmonic Functions and Norm Estimates for Inverses of Singular Integrals in Two Dimensions

In this paper, we establish sharp well-posedness results for tangential derivative problems for the Laplacian with data in L ^p , 1 < p < ∞, on curvilinear polygons. Furthermore, we produce norm estimates/formulas for inverses of singular integral operators relevant for the Dirichlet, Neumann, tangential derivative, and transmission boundary value problems associated with the Laplacian in a distinguished subclass of Lipschitz domains in two dimensions. Our approach relies on Calderón-Zygmund theory and Mellin transform techniques.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012