Ambrosetti Rabinowitz Type Condition Research Articles

On a class of generalized capillarity system involving fractional ψ$$ \psi $$‐Hilfer derivative with p(·)$$ p\left(\cdotp \right) $$‐Laplacian operator

This research delves into a comprehensive investigation of a class of ‐Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter in appropriate fractional ‐Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional ‐Hilfer generalized capillary phenomena.

Existence of solutions for Kirchhoff-double phase anisotropic variational problems with variable exponents

<p>This paper is devoted to dealing with a kind of new Kirchhoff-type problem in $ \mathbb{R}^N $ that involves a general double-phase variable exponent elliptic operator $ \mathit{\boldsymbol{\phi}} $. Specifically, the operator $ \mathit{\boldsymbol{\phi}} $ has behaviors like $ |\tau|^{q(x)-2}\tau $ if $ |\tau| $ is small and like $ |\tau|^{p(x)-2}\tau $ if $ |\tau| $ is large, where $ 1 < p(x) < q(x) < N $. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature.</p>

High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition

Abstract In this article, we study the following general Kirchhoff type equation: − M ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + u = a ( x ) f ( u ) in R 3 , -M\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+u=a\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where inf R + M > 0 {\inf }_{{{\mathbb{R}}}^{+}}M\gt 0 and f f is a superlinear subcritical term. By using the Pohozǎev manifold, we obtain the existence of high energy solutions of the aforementioned equation without the well-known Ambrosetti-Rabinowitz type condition.

Existence of nontrivial solutions for Schrodinger-Kirchhoff equations with indefinite potentials

We consider a class of Schrodinger-Kirchhoff equations in R3 with a general nonlinearity g and coercive sign-changing potential V so that the Schrodinger operator -aΔ +V is indefinite. The nonlinearity considered here satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μ G(t)>0 with μ>3. We obtain the existence of nontrivial solutions for this problem via Morse theory.

Existence and multiplicity of nontrivial solutions for a class of Kirchhoff-Schrödinger-Poisson systems

We consider a class of Kirchhoff-Schrödinger-Poisson systems. By using the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem of Rabinowitz, we prove the existence and multiplicity of nontrivial solutions. Our conditions weaken the Ambrosetti Rabinowitz type condition.

Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case

In present paper, we prove the existence of solutions (λ,u)∈R×H1(RN) to the following Schrödinger equation{−Δu(x)+V(x)u(x)+λu(x)=g(u(x))inRN0≤u(x)∈H1(RN),N≥3 satisfying the normalization constraint ∫RNu2dx=a. We treat the so-called mass super-critical case here. Under an explicit smallness assumption on V with lim|x|→∞⁡V(x)=supx∈RN⁡V(x) and some Ambrosetti-Rabinowitz type conditions on g, we can prove the existence of ground state normalized solutions for prescribed mass a>0. Furthermore, we emphasize that the mountain pass characterization of a minimizing solution of the probleminf⁡{∫[12|∇u|2+12V(x)u2−G(u)]dx:‖u‖L2(RN)2=a,P[u]=0}, where G(s)=∫0sg(τ)dτ andP[u]=∫[|∇u|2−12〈∇V(x),x〉u2−N(12g(u)u−G(u))]dx.

Existence Results for Double Phase Problem in Sobolev–Orlicz Spaces with Variable Exponents in Complete Manifold

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti–Rabinowitz type condition in the framework of Sobolev–Orlicz spaces with variable exponents in complete manifold. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder ine-quality, continuous and compact embedding results are proved.

Fractional double phase Robin problem involving variable order-exponents without Ambrosetti–Rabinowitz condition

We consider a fractional double phase Robin problem involving variable order and variable exponents. The nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz-type condition. By using a variational methods, we investigate the multiplicity of solutions.

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Quasilinear double phase problems in the whole space via perturbation methods

We are concerned with the following double phase problems in the whole space $$ \begin{aligned} -{\rm div}(|\nabla u|^{p-2}\nabla u + & \mu(x)|\nabla u|^{q-2}\nabla u) \\ & +|u|^{p-2}u+\mu(x)|u|^{q-2}u= f(x,u)\;{\rm in}\; \mathbb{R}^N. \end{aligned}$$ The nonlinearity is super-linear but does not satisfy the Ambrosetti-Rabinowitz type condition. The main difficulty is that weak limits of $(PS)$ sequences are not always weak solutions of the problem. To overcome this difficulty, we add {a} potential term and, using the mountain pass theorem, we get weak solutions $u_\lambda$ of the perturbed equations. First, we prove that $u_\lambda\rightharpoonup u$ as $\lambda\rightarrow 0$. Then, via a vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.

On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)$-Laplacian and without Ambrosetti-Rabinowitz type condition

On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)$-Laplacian and without Ambrosetti-Rabinowitz type condition

Multiplicity results for a system involving the p(x)- Laplacian operator

In this manuscript, it is considered existence and multiplicity of solutions for a system involving the -Laplacian operator which are related to several applications such as electrorheological fluids, image processing, robotics and space technology. A first solution for the system considered is obtained, under general conditions, by combining sub-supersolutions with a minimization argument in convex sets. Additionally, considering an Ambrosetti-Rabinowitz type condition and applying the Mountain Pass Theorem it is proved the existence of a second solution for the problem. The hypotheses allow to consider an wide class of nonlinearities and the results in this paper improve and complement some recent contributions.

Existence of ground state solutions for Kirchhoff-type problem with variable potential

The purpose of this paper is to study the following Kirchhoff-type equation where a>0 and b>0 are constants. Suppose that the nonnegative continuous potential V is not an asymptotic constant at infinity, and f satisfies some relatively weak conditions in the absence of the usual Ambrosetti–Rabinowitz type condition or monotonicity condition on . The result of this paper can be applied to the case where with . By using some new techniques and subtle analysis, we prove that the above problem admits at least one ground state solution. It is worth mention that our result generalize those obtained in Li and Ye [Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in . J Differ Equ. 2014;257:566–600], Tang and Chen [Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc Var Partial Differ Equ. 2017;56:110–134], Guo [Ground states for Kirchhoff equations without compact condition. J Differ Equ. 2015;259:2884–2902] and some other related literatures. In particular, we give a proof for the Pohožaev type identity associated with the above equation, when V is unbounded.

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations{Δ2u−Δu+V(x)u−12uΔ(u2)=g(u),inRN,u∈H2(RN) where Δ2=Δ(Δ) is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μG(t)≥0 with μ>3. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.

On nontrivial solutions of nonlinear Schr\xf6dinger equations with sign-changing potential

In this paper, we consider the superlinear Schrödinger equation with bounded potential well. The potential here is allowed to be sign-changing. Without assuming the Ambrosetti–Rabinowitz-type condition, we prove the existence of a nontrivial solution and multiplicity results.

Homoclinic Solutions of Nonlinear Laplacian Difference Equations Without Ambrosetti-Rabinowitz Condition

The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.

An Existence Result for a Class of p(x)—Anisotropic Type Equations

In this paper, we study a class of anisotropic variable exponent problems involving the p→(.)-Laplacian. By using the variational method as our main tool, we present a result regarding the existence of solutions without the so-called Ambrosetti–Rabinowitz-type conditions.

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t| p–2 t at infinity

Abstract We consider the following p-harmonic problem Δ ( | Δ u | p − 2 Δ u ) + m | u | p − 2 u = f ( x , u ) , x ∈ R N , u ∈ W 2 , p ( R N ) , $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2p ≥ 4 and lim t → ∞ f ( x , t ) | t | p − 2 t = l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ uniformly in x, which implies that f(x, t) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f(x, u) ≡ f(u), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

Ground state solutions for Choquard equations with Hardy potentials and critical nonlinearity

We are concerned with discussing the ground state solutions of the Choquard equation with the Hardy potentials and critical Sobolev exponent: where , , , is the Riesz potential, is the critical Sobolev exponent, and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor any monotonicity condition. Using some new variational and analytic techniques, we obtain a ground state solution of Pohoaev type for the given problem.

Infinitely many solutions for a generalized p(·)‐Laplace equation involving Leray–Lions type operators

We study the existence of infinitely many solutions for a generalized p(·)‐Laplace equation involving Leray–Lions operators. Firstly, under a p(·)‐sublinear condition for nonlinear term, we obtain a sequence of solutions approaching 0 by showing a new a priori bound for solutions. Secondly, for a p(·)‐superlinear condition, we produce a sequence of solutions whose Sobolev norms diverge to infinity when the nonlinear term satisfies a couple of generalized Ambrosetti–Rabinowitz type conditions in which each associated energy functional holds the Palais–Smale condition. Lastly, we deal with a case without the Ambrosetti–Rabinowitz type condition in which an associated energy functional holds the Cerami condition and establish a sequence of solutions whose Sobolev norms diverge to infinity.