By using a generalization of the Struwe-Jeanjean monotonicity trick we prove the existence of a non-trivial weak solution for the following problem
where
is a smooth bounded open set in
,
,
,
,
is ...the fractional p-Laplace operator and
is a continuous function with
and there are
and
such that
whenever
.
This paper is mainly related to multiple nontrivial solutions of the elliptic boundary value problem{−Δu=|u|p−2u+f(x,u),x∈Ω,u=0,x∈∂Ω, for p∈(2,2⁎). It is reasonable to guess that for dimΩ≥2 above ...problem possesses infinitely many distinct solutions since this is proved to be true for ODE. However, so far one does not even know if there exists a fourth nontrivial solution. By using a new homological linking theorem, Morse theory, and some precise estimates we disclose the relationship among the gaps of consecutive eigenvalues of Laplace operator, growth trend of nonlinear terms and the existence of multiple solutions of superlinear elliptic boundary value problem. Moreover, as p is close to 2, we get the fourth nontrivial solution under appropriate hypotheses, where f(x,u) satisfies Ambrosetti–Rabinowitz condition.
In this paper, we consider the existence of solutions for the quasilinear elliptic problem:(P1){−div(A(x,u)∇u)+12At(x,u)|∇u|2=g(x,u)+h(x),inΩ,u=0,on∂Ω, where Ω⊂RN is an open bounded domain, N≥3, the ...real term A(x,t), At(x,t)=∂A∂t(x,t) and g(x,t) satisfy Carathéodory condition on Ω×R and h:Ω→R is a given measurable function. The intention of the article is to get new results of the existence of infinitely many weak solutions of the problem by weaken the Ambrosetti and Rabinowitz condition. We use a variant of perturbation techniques introduced by Rabinowitz (1982) to overcome the lack of symmetry. This extends the previous results.
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.
We present an elementary proof of the existence of a nontrivial weak periodic solution for a nonlinear fractional problem driven by a relativistic Schrödinger operator with periodic boundary ...conditions and involving a periodic continuous subcritical nonlinearity satisfying a more general Ambrosetti-Rabinowitz condition.
We carry out an investigation of the existence of infinitely many solutions to a fractional
p
-Kirchhoff-type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet ...boundary condition. Further, the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A ‘
C
1
versus
W
0
s
,
p
’ analysis has also been discussed.
In this paper we study the nonlinear Schrödinger equation: {−Δu+V(x)u=f(x,u),u∊H1(RN). We give general conditions which assure the existence of ground state solutions. Under a Nehari type condition, ...we show that the standard Ambrosetti–Rabinowitz super-linear condition can be replaced by a more natural super-quadratic condition.
Dans cet article nous étudions l'équation non-linéaire de Schrödinger : {−Δu+V(x)u=f(x,u),u∊H1(RN). Nous donnons les conditions générales qui garantissent l'existence de solutions d'énergie minimale. Sous une condition de type Nehari, nous démontrons que la condition super-linéaire d'Ambrosetti–Rabinowitz peut être remplacée par une condition super-quadratique plus naturelle.
We study the quasilinear equation (P)−div(A(x,u)|∇u|p−2∇u)+1pAt(x,u)|∇u|p+|u|p−2u=g(x,u)inRN,with N≥3, p>1, where A(x,t), At(x,t)=∂A∂t(x,t) and g(x,t) are Carathéodory functions on RN×R.
Suitable ...assumptions on A(x,t) and g(x,t) set off the variational structure of (P) and its related functional J is C1 on the Banach space X=W1,p(RN)∩L∞(RN). In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of J restricted to Xr, subspace of the radial functions in X.
Following an approach which exploits the interaction between ‖⋅‖X and the norm on W1,p(RN), we prove the existence of at least one weak bounded radial solution of (P) by applying a generalized version of the Ambrosetti–Rabinowitz Mountain Pass Theorem.
PDF
