Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 729-766
Central China Normal University
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics
and Statistics
Wuhan, 430079, P.R. China; heyi19870113 'at' 163.com
Central China Normal University
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics
and Statistics
Wuhan, 430079, P.R. China; ligb 'at' mail.ccnu.edu.cn
Abstract. We are concerned with the following Schrödinger-Poisson equation with critical nonlinearity:
-ε2Δu +
V(x)u + ψu =
λ|u|p-2u + |u|4u
in R3,
-ε2Δψ
= u2 in R3, u > 0,
u ∈ H1(R3),
where ε > 0 is a small positive parameter, λ > 0, 3 < p ≤ 4. Under certain assumptions on the potential V, we construct a family of positive solutions uε ∈ H1(R3) which concentrates around a local minimum of V as ε → 0. Subcritical growth Schrödinger-Poisson equation
-ε2Δu +
V(x)u + ψu = f(u)
in R3,
-ε2Δψ
= u2 in R3, u > 0,
u ∈ H1(R3),
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 ∫0ug(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.
2010 Mathematics Subject Classification: Primary 35J20, 35J60, 35J92.
Key words: Existence, concentration, Schrödinger-Poisson equation, critical growth.
Reference to this article: Y. He and G. Li: Schrödinger-Poisson equations in R3 involving critical Sobolev exponents. Ann. Acad. Sci. Fenn. Math. 40 (2015), 729-766.
doi:10.5186/aasfm.2015.4041
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