The existence of solutions for an impulsive fractional coupled system of (p,q)‐Laplacian type without the Ambrosetti‐Rabinowitz condition

In this article, based on the variational approach, the existence of at least one nontrivial solution is studied for (p, q)‐Laplacian type impulsive fractional differential equations involving Riemann‐Liouville derivatives. Without the usual Ambrosetti‐Rabinowitz condition, the nonlinearity f in the paper is considered under some suitable assumptions.     

Nontrivial solutions of some fractional problems

In this paper, we study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. Furthermore, we give an extension of Ambrosetti–Rabinowitz condition, a non-Ambrosetti–Rabinowitz condition and apply to study the fractional Laplacian equation. We obtain some different existence results in this setting by using Fountain Theorem. Our results are extension of some problems studied by Bisci et al. (2016) and Binlin et al. (2015).     

Multiple homoclinic orbits for a class of the discrete p‐Laplacian with unbounded potentials

This paper is devoted to study the existence results of a sequence of infinitely many homoclinic orbits for the discrete p‐Laplacian with unbounded potentials without the Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem using the mountain pass theorem, the fountain theorem, and dual fountain theorem.     

Asymmetric (<Emphasis Type="Italic">p</Emphasis>, 2)-equations with double resonance

We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti–Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions.     

Fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian Equations with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition

The main purpose of this paper is to investigate the existence of nontrivial solutions to a class of quasilinear non-local problems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth (subcritical polynomial growth) at \(\infty \). Some existence results for nontrivial solution are obtained using mountain pass theorem combined with the fractional Moser–Trudinger inequality.     

A multiplicity theorem for <Emphasis Type="Italic">p</Emphasis>-superlinear <Emphasis Type="Italic">p</Emphasis>-Laplacian equations using critical groups and morse theory

We study a nonlinear elliptic equation driven by the Dirichlet p-Laplacian and with a Carathéodory nonlinearity. We assume that the nonlinearity exhibits a p-superlinear growth near infinity but need not satisfy the Ambrosetti–Rabinowitz condition. Using truncation techniques, minimax methods and Morse theory, we show that the problem admits at least three nontrivial solutions, two of which have constant sign (one positive, the other negative).     

Superlinear problems without Ambrosetti and Rabinowitz growth condition

Superlinear elliptic boundary value problems without Ambrosetti and Rabinowitz growth condition are considered. Existence of nontrivial solution result is established by combining some arguments used by Struwe and Tarantello and Schechter and Zou (also by Wang and Wei). Firstly, by using the mountain pass theorem due to Ambrosetti and Rabinowitz is constructed a solution for almost every parameter λ by varying the parameter λ. Then, it is considered the continuation of the solutions.     

A note on the Ambrosetti–Rabinowitz condition for an elliptic system

In this note, we study the existence and multiplicity of solutions for a system of coupled elliptic equations. We introduce a revised Ambrosetti–Rabinowitz condition, and show that the system has a nontrivial solution or even infinitely many solutions.     

Eigenvalue Problems for Hemivariational Inequalities

We consider a semilinear eigenvalue problem with a nonsmooth potential (hemivariational inequality). Using a nonsmooth analog of the local Ambrosetti–Rabinowitz condition (AR-condition), we show that the problem has a nontrivial smooth solution. In the scalar case, we show that we can relax the local AR-condition. Finally, for the resonant λ?=?λ ₁ problem, using the nonsmooth version of the local linking theorem, we show that the problem has at least two nontrivial solutions. Our approach is variational, using minimax methods from the nonsmooth critical point theory.     

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).     

On superlinear problems without the Ambrosetti and Rabinowitz condition

Existence and multiplicity results are obtained for superlinear p-Laplacian equations without the Ambrosetti and Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. Our results extend the recent results of Miyagaki and Souto [O. Miyagaki, M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008) 3628-3638].     

Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions

This paper deals with the existence of solutions for a class of p(x)-biharmonic equations with Navier boundary conditions. The approach is based on variational methods and critical point theory. Indeed, we investigate the existence of two solutions for the problem under some algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of the third solution for the problem.     

由p-Laplacian算子导出的半变分不等式的特征值问题

研究了一类具有非光滑局部Lipschitz位势(半变分不等式)的非线性特征值问题其中1相似文献   

Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity

We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory function which is (p?1)-superlinear but does not satisfy the Ambrosetti–Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann Inst H Poincaré Anal Non Linéaire 8(1):43–57, 1991). Subsequently, by imposing additional conditions on the nonlinearity \({f(x,\cdot)}\), we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of (p, 2)-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters \({\lambda > 0}\) sufficiently small where one solution vanishes in the Sobolev norm as \({\lambda \to 0^+}\) and the other one blows up (again in the Sobolev norm) as \({\lambda \to 0^+}\).     

Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.     

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.     

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.     

A multiplicity theorem for p-superlinear p-Laplacian equations using critical groups and morse theory

We study a nonlinear elliptic equation driven by the Dirichlet p-Laplacian and with a Carathéodory nonlinearity. We assume that the nonlinearity exhibits a p-superlinear growth near infinity but need not satisfy the Ambrosetti–Rabinowitz condition. Using truncation techniques, minimax methods and Morse theory, we show that the problem admits at least three nontrivial solutions, two of which have constant sign (one positive, the other negative).     

Existence of bound and ground states for a class of Kirchhoff‐Schrödinger equations involving critical Trudinger‐Moser growth

In this paper, we discuss the existence of bound and ground state solutions for a class of fractional Kirchhoff equations defined on the whole real line. The equation involves a nonlinear term with critical exponential growth in the Trudinger‐Moser sense. We deal with periodic and asymptotically periodic potential, which may change sign. We handle with the lack of compactness because of the unboundedness of the domain and the critical behavior of the nonlinearity. The main theorems are stated without the well‐known Ambrosetti‐Rabinowitz condition at infinity.     

Standing waves for 4‐superlinear Schrödinger‐Kirchhoff equations

We consider standing waves for 4‐superlinear Schrödinger–Kirchhoff equations in with potential indefinite in sign. The nonlinearity considered in this study satisfies a condition that is much weaker than the classical Ambrosetti–Rabinowitz condition. We obtain a nontrivial solution and, in the case of odd nonlinearity, an unbounded sequence of solutions via the Morse theory and the fountain theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.