2014
DOI: 10.1007/s00526-014-0736-7
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Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case

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Cited by 15 publications
(15 citation statements)
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“…However, while a suitable decay condition on a(x) − a ∞ appears quite reasonable, the second condition seems essentially due to technical motives. Hence, in subsequent papers some efforts have been made to drop this condition, but, until now, with successful results only in the planar case N = 2 and assuming polynomial decay of a(x) to a ∞ ( [15,17]). On the other hand, it is worth remarking that a careful analysis of the proofs in [12,[15][16][17]30] makes the reader understand that the symmetry in [16,30], the small oscillation assumption in [12], the dimension restriction N = 2 in [15,17], in spite of the different arguments and methods displayed in the papers, are essentially related to the same basic fact for the proof: working with functions having bumps located in regions where a(x) − a ∞ is small.…”
Section: Introduction and Statement Of The Main Resultsmentioning
confidence: 99%
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“…However, while a suitable decay condition on a(x) − a ∞ appears quite reasonable, the second condition seems essentially due to technical motives. Hence, in subsequent papers some efforts have been made to drop this condition, but, until now, with successful results only in the planar case N = 2 and assuming polynomial decay of a(x) to a ∞ ( [15,17]). On the other hand, it is worth remarking that a careful analysis of the proofs in [12,[15][16][17]30] makes the reader understand that the symmetry in [16,30], the small oscillation assumption in [12], the dimension restriction N = 2 in [15,17], in spite of the different arguments and methods displayed in the papers, are essentially related to the same basic fact for the proof: working with functions having bumps located in regions where a(x) − a ∞ is small.…”
Section: Introduction and Statement Of The Main Resultsmentioning
confidence: 99%
“…Hence, in subsequent papers some efforts have been made to drop this condition, but, until now, with successful results only in the planar case N = 2 and assuming polynomial decay of a(x) to a ∞ ( [15,17]). On the other hand, it is worth remarking that a careful analysis of the proofs in [12,[15][16][17]30] makes the reader understand that the symmetry in [16,30], the small oscillation assumption in [12], the dimension restriction N = 2 in [15,17], in spite of the different arguments and methods displayed in the papers, are essentially related to the same basic fact for the proof: working with functions having bumps located in regions where a(x) − a ∞ is small. This observation is, in a way, also validated by the results of [11] where the existence of infinitely many positive and infinitely many nodal multi-bump solutions to (1.1) is shown considering potentials, having slow decay but not small oscillation neither symmetry, which are asked to sink in some large regions of R N to the end of localizing the bumps suitably far and, when one looks for changing sing solutions, to control the attractive effect of positive and negative bumps each other.…”
Section: Introduction and Statement Of The Main Resultsmentioning
confidence: 99%
“…In this regard, there are two recent papers with different approaches. In , del Pino, the second author and Yao used the intermediate Lyapunov–Schmidt reduction method to prove the existence of infinitely many positive solutions to for non‐symmetric potentials, when N=2, and (m,p,σ) satisfies min1,p12m>2,σ>2.On the other hand, Devillanova and Solimini used variational methods to show that there are infinitely many positive solutions to for non‐symmetric potentials, when N=2, and V(x) satisfies A1|x|sV(x)VA2|x|,forxlargeands<4.Moreover, if V(x) tends to V from above with a suitable V(x)V,trueprefixlim|x|(V(x)V)eη|x|=+forsomeη0,Vand V satisfies a global condition: trueprefixsupxdouble-struckRNV(x)V…”
Section: Introductionmentioning
confidence: 99%
“…Here we allow N3 and m4 (comparing with ). Our result suggests that the following conjecture should be true: Corollary There are infinitely many positive solutions to provided V satisfies…”
Section: Introductionmentioning
confidence: 99%
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