Ambrosetti Rabinowitz Condition Research Articles

Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator

In this paper, we study the multiplicity of weak solutions to the boundary value problem where Ω is a bounded domain with smooth boundary in is odd in ξ and is a perturbation term. Under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. Here we do not require that f satisfies the Ambrosetti-Rabinowitz (AR) condition. The conditions on f and g are relatively weak and our result is new even in the case , i.e. for the classical Laplace equation with the Dirichlet boundary condition.

Singular Trudinger–Moser inequality and fractional [formula omitted]-Laplace equations in [formula omitted

In this paper, we establish a singular Trudinger–Moser inequality in RN. Applying that result, we study the existence of solution to N∕s-fractional Laplacian equation LN∕ssu(x)+V(x)|u|Ns−2u=f(x,u)|x|γ,where γ∈[0,N), 0<s<1,LN∕ss is a nonlocal fractional operator with singular kernel K:RN∖{0}→R+,f is a continuous function on RN×R which does not satisfy the Ambrosetti–Rabinowitz condition. By using Mountain Pass Theorem, we obtain the existence of solution to above equation. Furthermore, when f satisfies the Ambrosetti–Rabinowitz condition, we obtain the existence to solution of that equation with negative energy. In our best knowledge, it is the first time the existence to solution of above equation is studied.

Existence and multiplicity of solutions for Dirichlet problem of p(x)-Laplacian type without the Ambrosetti-Rabinowitz condition

This paper deals with the following variational problem with variable exponents{−div((1+|∇u|p(x)1+|∇u|2p(x))|∇u|p(x)−2∇u)=f(x,u),inΩ,u=0,on∂Ω, where Ω⊆RN (N≥2) be a bounded domain with the smooth boundary ∂Ω, p:Ω‾→R is a Lipschitz continuous function with 1<p−:=essinfx∈Ω⁡p(x)≤p(x)≤p+:=esssupx∈Ω⁡p(x)<N and f∈C(Ω×R,R) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.

Positive and nodal solutions for nonlinear nonhomogeneous parametric Neumann problems

We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda&gt;0$$ We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution. Finally using the extremal constant sign solutions we produce a smooth nodal solution.

p -biharmonic equation with Hardy–Sobolev exponent and without the Ambrosetti–Rabinowitz condition

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="p"> <mml:mi>p</mml:mi> </mml:math>-biharmonic equation with Hardy–Sobolev exponent and without the Ambrosetti–Rabinowitz condition

Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by $ (\Phi_{1}, \Phi_{2}) $-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.

Schr&amp;#246;dinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions

We consider the following Schrodinger-Poisson system $ \left\{ \begin{array}{l} {\rm{ - }}\Delta u + V\left(x \right)u + \phi u = \lambda f\left(u \right)\; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}, \\ - \Delta \phi = {u^2}, \mathop {\lim }\limits_{|x| \to + \infty } \phi = 0, \; \; \; \; \; \; \; \; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}. \end{array} \right. $ Unlike most other papers on this problem, the Schrodinger-Poisson system without any growth and Ambrosetti-Rabinowitz condition is considered in this paper. Firstly, by Jeanjean's monotonicity trick and the mountain pass theorem, we prove that the problem possesses a positive solution for large value of $\lambda$. Secondly, we establish the multiplicity of solutions via the symmetric mountain pass theorem.

Parameter dependence for the positive solutions of nonlinear, nonhomogeneous Robin problems

We consider a parametric nonlinear Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential. The reaction term is $$(p-1)$$-superlinear but need not satisfy the usual Ambrosetti–Rabinowitz condition. We look for positive solutions and prove a bifurcation-type result for the set of positive solutions as the parameter $$\lambda >0$$ varies. Also we prove the existence of a minimal positive solution $$u_\lambda ^*$$ and determine the monotonicity and continuity properties of the map $$\lambda \rightarrow u_\lambda ^*$$.

Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with -Laplacian

AbstractWe are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic-Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at least three weak solutions for the problem in a special case.

Weak solutions for a (p(z),q(z))-Laplacian Dirichlet problem

We establish the existence of a nontrivial and nonnegative solution for a double phase Dirichlet problem driven by a (p(z); q(z))-Laplacian operator plus a potential term. Our approach is variational, but the reaction term f need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition.

A multiplicity theorem for parametric superlinear (p,q)-equations

We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.

Existence Result for Fractional Klein-Gordon-Maxwell System with Quasicritical Potential Vanishing at Infinity

The following fractional Klein-Gordon-Maxwell system is studied (-Δ)p stands for the fractional Laplacian, ω > 0 is a constant, V is vanishing potential and K is a smooth function. Under some suitable conditions on K and f, we obtain a Palais-Smale sequence by using a weaker Ambrosetti-Rabinowitz condition and prove the ground state solution for this system by employing variational methods. In particular, this kind of problem is a vast range of applications and challenges.

Existence of solutions for fractional $p$-Kirchhoff type equations with a generalized Choquard nonlinearity

In this article, we establish the existence of solutions to the fractional $p$-Kirchhoff type equations with a generalized Choquard nonlinearity without assuming the Ambrosetti-Rabinowitz condition.

Nonlinear nonhomogeneous singular problems

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a $(p-1)$-superlinear perturbation which does not satisfy the usual Ambrosetti-Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a "bifurcation-type" theorem that describes the set of positive solutions as the parameter $\lambda$ moves on the positive semiaxis. We also show that for every $\lambda>0$, the problem has a smallest positive solution $u^*_\lambda$ and we demonstrate the monotonicity and continuity properties of the map $\lambda\mapsto u^*_\lambda$.

Constant sign and nodal solutions for superlinear double phase problems

Abstract We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

Schrödinger–Kirchhoff–Hardy &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;–fractional equations without the Ambrosetti–Rabinowitz condition

This paper is devoted to the study of the following Schrödinger–Kirchhoff–Hardy equation in $ \mathbb R^n $ \begin{document}$ M\left(\iint_{\mathbb R^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\right)(-\Delta)^{s}_pu+V(x)|u|^{p-2}u-\mu\frac{|u|^{p-2}u}{|x|^{ps}} = f(x, u), $\end{document} where $ (-\Delta)^s_p $ is the fractional $ p $–Laplacian, with $ s\in(0, 1) $ and $ p&gt;1 $, dimension $ n&gt;ps $, $ M $ models a Kirchhoff coefficient, $ V $ is a positive potential, $ f $ is a continuous nonlinearity and $ \mu $ is a real parameter. The main feature of the paper is the combination of a Kirchhoff coefficient and a Hardy term with a suitable function $ f $ which does not necessarily satisfy the Ambrosetti–Rabinowitz condition. Under different assumptions for $ f $ and restrictions for $ \mu $, we provide existence and multiplicity results by variational methods.

Fractional Choquard Equations with Confining Potential With or Without Subcritical Perturbations

Abstract In this paper, we consider fractional Choquard equations with confining potentials. First, we show that they admit a positive ground state and infinitely many bound states. Then we prove the existence of two signed solutions when a superlinear and subcritical perturbation is added; in this case, the main feature is that such a perturbation does not satisfy the usual Ambrosetti–Rabinowitz condition.

Solutions and positive solutions for superlinear Robin problems

We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.

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).

Two homoclinic orbits for some second-order Hamiltonian systems

This paper is concerned with the existence of homoclinic orbits for a class of second order Hamiltonian systems considering a non-periodic potential and a weaker Ambrosetti-Rabinowitz condition. By considering an auxiliary problem, we show the existence of two different approximative sequences of periodic solutions, the first one of mountain pass type and the second one of local minima. We obtain two different homoclinic orbits by passing to the limit in such sequences. As a relevant application, we obtain another homoclinic solution for the Hamiltonian system studied in \cite{IJ}.