2015
DOI: 10.1142/s021919971450045x
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Multiple positive solutions for a class of concave–convex elliptic problems in ℝNinvolving sign-changing weight, II

Abstract: In this paper, we study the following concave–convex elliptic problems: [Formula: see text] where N ≥ 3, 1 < q < 2 < p < 2* = 2N/(N - 2), λ > 0 and μ < 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of fλ(x) and gμ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded d… Show more

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Cited by 24 publications
(15 citation statements)
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“…Since (P λ1 ) has no solution, it follows from the strong maximum principle that u 0 = 0 in H 1 0 (B R ). Note that α n ↑ λ 1 , we can see from similar arguments used in the proof of [12,Lemma 5.2] that (iii) 0 < λ < aλ 1 and µ > bS 2 in the case N = 4;…”
Section: 3mentioning
confidence: 56%
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“…Since (P λ1 ) has no solution, it follows from the strong maximum principle that u 0 = 0 in H 1 0 (B R ). Note that α n ↑ λ 1 , we can see from similar arguments used in the proof of [12,Lemma 5.2] that (iii) 0 < λ < aλ 1 and µ > bS 2 in the case N = 4;…”
Section: 3mentioning
confidence: 56%
“…Furthermore, a similar argument used in the proof of [12,Lemma 5.2] shows that m α < m 0 for all α ∈ (0, λ 1 ). It follows from the Hölder inequality that…”
Section: The Case Of 2 = Q < P < 2 *mentioning
confidence: 76%
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“…By the variational method, they obtained the existence and multiplicity of positive solutions to the above problem. Subsequently, an increasing number of researchers have paid attention to semilinear elliptic equations with critical exponent and concave-convex nonlinearities; for example, see [ 1 , 5 , 13 , 14 , 27 , 29 ] and the references therein.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…Here we cite the pioneer work [2] where several results are proved on bounded domains Ω ⊂ R N . In the whole space concave-convex nonlinearities have been considered assuming extra assumptions on the potential V , see [8,13,34]. Another contribution in this work is to consider the nonlinear Rayleigh quotient proving existence of a parameter λ * > 0 such that Problem (1.1) admits at least two solutions for each λ ∈ (0, λ * ].…”
mentioning
confidence: 99%