Standing Waves for a Class of Schrödinger–poisson Equations in R Involving Critical Sobolev Exponents
نویسندگان
چکیده
−ε2∆u + V (x)u + ψu = f(u) in R, −ε2∆ψ = u in R, u > 0, u ∈ H(R), has been studied extensively, where the assumption for f(u) is that f(u) ∼ |u|p−2u with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz condition which forces the boundedness of any Palais– Smale sequence of the corresponding energy functional of the equation. The more difficult critical case is studied in this paper. As g(u) := λ|u|p−2u + |u|4u with 3 < p ≤ 4 does not satisfy the Ambrosetti–Rabinowitz condition (∃μ > 4, 0 < μ ́ u 0 g(s) ds ≤ g(u)u), the boundedness of Palais– Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function g(s) s3 is not increasing for s > 0 prevents us from using the Nehari manifold directly as usual. The main result we obtained in this paper is new.
منابع مشابه
Quasilinear Schrödinger equations involving critical exponents in $mathbb{textbf{R}}^2$
We study the existence of soliton solutions for a class of quasilinear elliptic equation in $mathbb{textbf{R}}^2$ with critical exponential growth. This model has been proposed in the self-channeling of a high-power ultra short laser in matter.
متن کاملThe Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent
In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.
متن کاملNonexistence results for a class of nonlinear elliptic equations involving critical Sobolev exponents
متن کامل
Multiple Positive Solutions for Degenerate Elliptic Equations with Critical Cone Sobolev Exponents on Singular Manifolds
In this article, we show the existence of multiple positive solutions to a class of degenerate elliptic equations involving critical cone Sobolev exponent and sign-changing weight function on singular manifolds with the help of category theory and the Nehari manifold method.
متن کاملON QUASILINEAR ELLIPTIC SYSTEMS INVOLVING MULTIPLE CRITICAL EXPONENTS
In this paper, we consider the existence of a non-trivial weaksolution to a quasilinear elliptic system involving critical Hardyexponents. The main issue of the paper is to understand thebehavior of these Palais-Smale sequences. Indeed, the principaldifficulty here is that there is an asymptotic competition betweenthe energy functional carried by the critical nonlinearities. Thenby the variatio...
متن کامل