CorrectMyText Users
English language
The text to correct from user nsak23
In this paper we would like to extend the notion of resonance to the (p; q) -Laplacian operator. Namely,
we will study the equation (1)
The term resonance refers to the case in which lambda is a principal eigenvalue of the problem (2)
For the case p = q > 1 the spectral properties of 2 are being established. It is known that in this
case there exists a smallest eigenvalue, i. e. , the principal eigenvalue 1, which is simple and isolated.
Also, the properties of the next smallest eigenvalue, 2, have been investigated in [2]. Beyond this, by
using Ljusternik-Schnirelmann theory, it was shown the existence of a no decreasing positive sequence
of eigenvalues 0 < 1 <::: < n <::: (see [8, 11, 6]). For the case p 6= q we have shown in [3] that 2
has a continuous family of positive eigenvalues.
Resonance problems around an eigenvalue of the p-Laplacian have been treated just recently in [1]
among others. The main tool in studding such problem is the use of variational approaches like the
Mountain Pass Theorem due to Ambrosetti-Rabinowitz, the Ekeland variational principal and the
Saddle point Theorem. For example, a multiple existence result is obtained in [1] by combing the two rst theorems. Neverthless, this approach is not applicable to our case because we can not decompose
the functional space to a direct sum of a nite dimension space and its orthogonal complement since
we can not ensure the nite multiplicity of the principal eigenvalue. Instead of this, We will apply a
weak version of the monotone operators theorem due to Leray-Lions to investigate problem 1.
Language: English
Language Skills: Native speaker, Proficiency
we will study the equation (1)
The term resonance refers to the case in which lambda is a principal eigenvalue of the problem (2)
For the case p = q > 1 the spectral properties of 2 are being established. It is known that in this
case there exists a smallest eigenvalue, i. e. , the principal eigenvalue 1, which is simple and isolated.
Also, the properties of the next smallest eigenvalue, 2, have been investigated in [2]. Beyond this, by
using Ljusternik-Schnirelmann theory, it was shown the existence of a no decreasing positive sequence
of eigenvalues 0 < 1 <::: < n <::: (see [8, 11, 6]). For the case p 6= q we have shown in [3] that 2
has a continuous family of positive eigenvalues.
Resonance problems around an eigenvalue of the p-Laplacian have been treated just recently in [1]
among others. The main tool in studding such problem is the use of variational approaches like the
Mountain Pass Theorem due to Ambrosetti-Rabinowitz, the Ekeland variational principal and the
Saddle point Theorem. For example, a multiple existence result is obtained in [1] by combing the two rst theorems. Neverthless, this approach is not applicable to our case because we can not decompose
the functional space to a direct sum of a nite dimension space and its orthogonal complement since
we can not ensure the nite multiplicity of the principal eigenvalue. Instead of this, We will apply a
weak version of the monotone operators theorem due to Leray-Lions to investigate problem 1.
Login or register to correct this text!
You can correct this text if you set follow language skills of English in settings: Native speaker, Proficiency
You can correct this text if you set follow language skills of English in settings: Native speaker, Proficiency
Please correct texts in English:
Please help with translation:
Arabic Armenian Azerbaijan Basque Belorussian Bosnian Bulgarian Catalan Chinese Croatian Czech Danish Dutch English Esperanto Estonian Finnish French Georgian German Greek Gujarati Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Javanese Kazakh Korean Kyrgyz Latin Latvian Macedonian Malay Mongolian Norwegian (bokmal) Norwegian (Nynorsk) Persian Polish Portuguese Romanian Russian Sanskrit Serbo-Croatian Slovak Spanish Swedish Tatar Thai Turkish Ukrainian Uzbek Vietnamese Welsh