Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in R N
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RESEARCH
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Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in RN Huei-li Lin* *
Correspondence: [email protected] Department of Natural Sciences in the Center for General Education, Chang Gung University, Tao-Yuan, 333, Taiwan, R.O.C.
Abstract In this paper, we investigate the effect of the coefficient f (x) of the subcritical nonlinearity. Under some assumptions, for sufficiently small ε , λ, μ > 0, there are at least k (≥ 1) positive solutions of the semilinear elliptic systems ⎧ α 2 q–2 α –2 β in RN ; ⎪ ⎨–ε u + u = λg(x)|u| u + α +β f (x)|u| u|v| –ε 2 v + v = μh(x)|v|q–2 v + αβ+β f (x)|u|α |v|β –2 v in RN ; ⎪ ⎩ u, v ∈ H1 (RN ), where α > 1, β > 1, 2 < q < p = α + β < 2∗ = 2N/(N – 2) for N ≥ 3. MSC: 35J20; 35J25; 35J65 Keywords: semilinear elliptic systems; subcritical exponents; Nehari manifold
1 Introduction For N ≥ , α > , β > and < q < p = α + β < ∗ = N/(N – ), we consider the semilinear elliptic systems ⎧ q– ⎪ ⎪ ⎨–ε u + u = λg(x)|u| u + ⎪ ⎪ ⎩
–ε v + v = μh(x)|v|q– v +
u > ,
α f (x)|u|α– u|v|β α+β β f (x)|u|α |v|β– v α+β
in RN ; in RN ;
(Eε,λ,μ )
v > ,
where ε, λ, μ > . Let f , g and h satisfy the following conditions: (A) f is a positive continuous function in RN and lim|x|→∞ f (x) = f∞ > . (A) there exist k points a , a , . . . , ak in RN such that f ai = max f (x) = for ≤ i ≤ k, x∈RN
and f∞ < . (A) g, h ∈ Lm (RN ) ∩ L∞ (RN ) where m = (α + β)/(α + β – q), and g, h . © 2012 Lin; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lin Boundary Value Problems 2012, 2012:118 http://www.boundaryvalueproblems.com/content/2012/1/118
Page 2 of 17
In [], if is a smooth and bounded domain in RN (N ≤ ), they considered the following system: ⎧ ⎪ ⎪ ⎨ε u – λ u = μ u + βuv ⎪ ⎪ ⎩
in ;
ε v – λ v = μ v + βu v
u > ,
in ;
v > ,
and proved the existence of a least energy solution in for sufficiently small ε > and β ∈ (–∞, β ). Lin and Wei also showed that this system has a least energy solution in RN for ε = and β ∈ (, β ). In this paper, we study the effect of f (z) of (Eε,λ,μ ). Recently, many authors [–] considered the elliptic systems with subcritical or critical exponents, and they proved the existence of a least energy positive solution or the existence of at least two positive solutions for these problems. In this paper, we construct the k compact PalaisSmale sequences which are suitably localized in correspondence of k maximum points of f . Then we could show that under some assumptions (A)-(A), for sufficiently small ε, λ, μ > , there are at least k (≥ ) positive solutions of the elliptic system (Eε,λ,μ ). By the change of variables x = εz,
u(z) = u(εz)
and
v(z) = v(εz),
System (Eε,λ,μ ) is transformed to ⎧ q– ⎪ ⎪–u + u
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