Ambrosetti Rabinowitz Type Condition Research Articles

Existence of solutions for a class of $p(x)$-curl systems arising in electromagnetism without (A-R) type conditions

In this paper, we study the existence and multiplicity of solutions for a class of of $p(x)$-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the $p(x)$-curl operator in [4, 15].

Existence of solutions for the double phase variational problems without AR-condition

We consider the following double phase problem: $$\begin{aligned} -\mathrm {div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)+V(x)|u|^{\alpha -2}u=f(x,u),\quad \mathrm {in}\ \mathbb {R}^{N}, \end{aligned}$$ where $$N\ge 2,$$ and $$\frac{q}{p}<1+\frac{1}{N}, a:\mathbb {R}^{N}\rightarrow [0,+\infty )~\mathrm {is~Lipschitz~continuous.}$$ The Ambrosetti–Rabinowitz type condition, that is so-called (AR) condition: there exist $$L>0,\theta >q$$ such that for $$|t|\ge L$$ and a.e. $$x\in \mathbb {R}^{N}$$ , $$\begin{aligned} 0<\theta F(x,t)\le t f(x,t), \end{aligned}$$ as well as the monotonicity of $$f(x,t)/|t|^{q-1}$$ are not assumed. Under appropriate assumptions on V and f, we prove that the above problem has at least a nontrivial solution.

Existence and concentration of ground state solutions for a class of Kirchhoff-type problems

This paper is concerned with the following singularly perturbed Kirchhoff-type problem −ε2a+εb∫R3|∇u|2dx△u+V(x)u=f(u),x∈R3;u∈H1(R3),where ε>0 is a small parameter, a,b>0 are two constants, V∈C(R3,R), and f∈C(R,R) is of super-linear growth at infinity and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor monotonicity condition on f(u)∕u3. By using some new techniques and subtle analyses, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0], the above problem has a ground state solution concentrating around global minimum of V in the semi-classical limit. Our results are available to the case that f(u)∼|u|s−2u for s∈(2,6), and extend the existing results concerning the case that f(u)∼|u|s−2u for s∈[4,6).

Standing waves with a critical frequency for a quasilinear Schrödinger equation

We consider the following quasilinear Schrödinger equation [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] satisfies a weaker growth condition than the Ambrosetti–Rabinowitz type condition in Byeon and Wang [Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 165(4) (2002) 295–316; Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. 18(2) (2003) 207–219]. We obtain the existence of the localized bound state solutions concentrating at an isolated component of the local minimum of [Formula: see text] and whose amplitude goes to 0 as [Formula: see text].

On a class of double-phase problem without Ambrosetti–Rabinowitz-type conditions

ABSTRACT We study the following double-phase problem where and 1<p<q<N. The aim is to determine the precise positive interval of λ for which the problem admits at least two nontrivial solutions via variational methods for the above equation. The primitive of the nonlinearity f is of super-q growth near infinity in u and allowed to be sign-changing. Furthermore, our assumptions are suitable and different from those studied previously.

Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations

In this paper, we study the following singularly perturbed Schrödinger-Poisson system{−ε2△u+V(x)u+ϕu=f(u)+u5,x∈R3,−ε2△ϕ=u2,x∈R3, where ε is a small positive parameter, V∈C(R3,R) and f∈C(R,R) satisfies neither the usual Ambrosetti-Rabinowitz type condition nor any monotonicity condition on f(u)/u3. By using some new techniques and subtle analysis, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0] the above system admits a semiclassical ground state solution vˆε with exponential decay at infinity. We also study the asymptotic behavior of {vˆε} as ε→0. In particular, our results can be applied to the nonlinearity f(u)∼|u|q−2u for q∈[3,4], and extend the previous work that only deals with the case in which f(u)∼|u|q−2u for q∈(4,6).

Superlinear Elliptic Equations with Variable Exponent via Perturbation Method

We are concerned with the following $p(x)$-Laplacian equations in $\mathbb{R}^{N}$$$ -\triangle _{p(x)} u+|u|^{p(x)-2}u= f(x,u)\quad \mbox{in } \mathbb{R} ^{N}. $$ The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution $u_{\lambda }$ of perturbation equations. First, we prove that $u_{\lambda }\rightharpoonup u$ as $\lambda \rightarrow 0$. Second, by using vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.

On a class of superlinear nonlocal fractional problems without Ambrosetti–Rabinowitz type conditions

On a class of superlinear nonlocal fractional problems without Ambrosetti–Rabinowitz type conditions

Multiple Solutions to a Caffarelli–Kohn–Nirenberg-Type Equation in Finsler Manifold

This paper deals with the existence of infinitely many solutions for a generalized form of Caffarelli–Kohn–Nirenberg-type equation in a Finsler manifold, with asymptotically linear term at infinity. This kind of nonlinear term creates some difficulties in the application of the symmetric mountain pass lemma because of the lack of Ambrosetti–Rabinowitz-type condition; hence, it is difficult to verify the classical (P S)c condition. To overcome this difficulty, we use an equivalent version of Cerami’s condition (C)c.

Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

AbstractWe are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

Ground state solutions for a class of Choquard equations with potential vanishing at infinity

This paper is dedicated to studying the following nonlinear Choquard equation−△u+V(x)u=(∫RNQ(y)F(u(y))|x−y|μdy)Q(x)f(u),u∈D1,2(RN), where N≥3, μ∈(0,N), V∈C(RN,[0,∞)), Q∈C(RN,(0,∞)), f∈C(R,R) and F(t)=∫0tf(s)ds. By combining the non-Nehari manifold approach with some new inequalities, we prove that the above equation has a ground state solution in the case when V(x)→0 as |x|→∞, where the strict monotonicity condition on f is not required. Moreover, by using perturbation method, we obtain the existence of a least energy solution in the zero mass case, i.e. V=0, where f satisfies the condition that F(t0)≠0 for some t0∈R instead of the usual Ambrosetti–Rabinowitz type condition. These results extend the ones in Alves, Figueiredo and Yang (2015) [1] and some related literature.

On existence solution for Schrödinger–Kirchhoff-type equations involving the fractional p-Laplacian in ℝN

ABSTRACTThe aim of this paper is to study the existence solution for Schrödinger–Kirchhoff-type equations involving nonlocal p-fractional Laplacianwhere is a real positive parameter, is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with f is a Carathéodory function on satisfying the Ambrosetti–Rabinowitz-type condition. Using Mountain Pass Theorem, we obtain the existence of above equations. Our result is a extension the problem studied by Pucci–Xiang–Zhang [1].

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

This paper is devoted to study the following Schrödinger-Poisson system where λ is a positive parameter, a ∈ C(R3,R+) has a bounded potential well Ω = a−1(0), b ∈ C(R3, R) is allowed to be sign-changing, K ∈ C(R3, R+) and f ∈ C(R, R). Without the monotonicity of f(t)=/|t|3 and the Ambrosetti-Rabinowitz type condition, we establish the existence and exponential decay of positive multi-bump solutions of the above system for , and obtain the concentration of a family of solutions as λ →+∞, where is determined by terms of a, b, K and f. Our results improve and generalize the ones obtained by C. O. Alves, M. B. Yang [3] and X. Zhang, S. W. Ma [38].

Variational analysis of anisotropic Schrödinger equations without Ambrosetti–Rabinowitz-type condition

This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schrodinger-type equations of the form $$\begin{aligned} \left\{ \begin{array}{ll} - \displaystyle \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x)|u|^{P^+_+-2}u =\lambda f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array}\right. \end{aligned}$$ without the Ambrosetti–Rabinowitz growth condition. Our arguments rely on the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter.

Infinitely many solutions for semilinear Δλ-Laplace equations with sign-changing potential and nonlinearity

In this paper, we prove the existence of infinitely many solutions for the following class of boundary value elliptic problems where Ω is a bounded domain in RN (N ≥ 2), Δλ is a strongly degenerate elliptic operator, V (x) is allowing to be sign-changing and f is a function with a more general super-quadratic growth, which is weaker than the Ambrosetti-Rabinowitz type condition.

Existence result for fractional Schrödinger–Poisson systems involving a Bessel operator without Ambrosetti–Rabinowitz condition

The present study is concerned with the nontrivial solutions for fractional Schrödinger–Poisson system with the Bessel operator. Under certain assumptions on the nonlinearity f, a nontrivial nonnegative solution is obtained by perturbation method for the given problem. In particular, the Ambrosetti–Rabinowitz type condition or the monotone assumption on the nonlinearity is unnecessary.

Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension

We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential V. Under suitable assumptions on V, using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti–Rabinowitz type condition or monotonicity assumptions.

A critical Kirchhoff type problem involving the fractional Laplacian in

In this paper, we are concerned with the existence of solutions for a critical p-Kirchhoff type problem driven by a nonlocal integro-differential operator:where is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with , , f is a Carathéodory function on satisfying the Ambrosetti–Rabinowitz type condition. Under some suitable assumptions, we obtain the existence of nontrivial solutions for above problem by applying the mountain pass theorem. A distinguished feature of this paper is that M(0) may be zero, which means that the problem is degenerate. Consequently, the main theorem extends in several directions the recent results of Autuori, Fiscella and Pucci [Nonlinear Anal. 2015;125:699–714].

Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions

In this paper we establish, using variational methods, the existence and multiplicity of weak solutions for a general class of quasilinear problems involving $$p(\cdot )$$ -Laplace type operators, with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz (A-R) type growth conditions, namely $$\begin{aligned} \left\{ \begin{array}{rcll} -{\text {div}}(a(|\nabla u|^{p(x)})|\nabla u|^{p(x)-2}\nabla u)&{}=&{}\lambda f(x,u) &{} \text{ in } \Omega ,\\ u&{}=&{}0 &{} \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$ By different types of versions of the Mountain Pass Theorem with Cerami condition, as well as, the Fountain and Dual Theorem with Cerami condition, we obtain some existence of weak solutions for the above problem under some considerations. Moreover, we show that the problem treated has at least one nontrivial solution for any parameter $$\lambda >0$$ small enough, and also that the solution blows up, in the Sobolev norm, as $$\lambda \rightarrow 0^{+}.$$ Finally, by imposing additional hypotheses on the nonlinearity $$f(x,\cdot ),$$ we get the existence of infinitely many weak solutions by using the Genus Theory introduced by Krasnoselskii.

Infinitely many solutions for a class of stationary Schrödinger equations with non-standard growth

In this paper, we study the existence of infinitely many solutions for a class of stationary Schrödinger type equations in involving the p(x)-Laplacian. The non-linearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. The main arguments are based on the geometry supplied by Fountain Theorem. We also establish a Bartsch type compact embedding theorem for variable exponent spaces.