Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space

By Y. Yang. 2011

165 citations recorded by the source corpus.

Abstract In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger–Moser type inequality. Under some assumptions on the potential and the nonlinearity, it is proved that there is a nontrivial positive weak solution to this equation. Also it is shown that there are two distinct positive weak solutions to a perturbation of the equation. The method of proving these results is combining Trudinger–Moser type inequality, Mountain-pass theorem and Ekelandʼs variational principle.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0b889315ce62ac39330fb198b90c337482009b16

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In Applied Mathematics

1. Federica Sani
2. Minbo Yang
3. Daniele Cassani
4. Nader Masmoudi
5. Xiaobao Zhu
6. Cristina Tarsi
7. Yong Lin
8. Y. Yang

In Computational Theory and Mathematics

1. Federica Sani
2. Minbo Yang
3. Xiaobao Zhu
4. Nader Masmoudi
5. Daniele Cassani
6. Alexander Grigorʼyan
7. Yong Lin
8. Y. Yang

Citations in the corpus

• Existence of positive solutions to some nonlinear equations on locally finite graphs (2016, unknown): “rthi [1], Adimurthi-Yadava[2], Adimuthi-Yang [3], Alves-Figueiredo [5], Cao [7], Ruf et al [8, 9], Ding-Ni [10], doO et al [11, 12, 13, 14], Jeanjean [16], Kryszewski-´ Szulkin [17], Panda [18], Yang [19, 20], and the referencestherein. For the Riemannian manifold case, we refer the reader to [15, 2”
• Trudinger-Moser Inequalities with the Exact Growth Condition in ℝ N and Applications (2015, background): “The authors of [4, 5, 13, 14, 16, 17, 20, 30], considering a potential V satisfying (1.”
• Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two (2016, background): “Such singular Trudinger-Moser inequalities are very important in the study of partial differential equations, see for examples [2, 4, 12, 30, 31].”
• Yamabe type equations on graphs (2016, background): “For related works, we refer the reader to [10,2,25,26] and the references there in.”
• Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in $\R^2$ (2016, background): “u|2 ≤ M < ∞, and α < α0 = 4π, then there exists a constant C, which depends only on M and α, such that Z R2 eα|u|2 −1 ≤ C(M,α). (1.6) TM2 We refer the reader to [3, 30] for related problems and [13, 31, 52] for recent advances on this topic. Actually just a few papers deal with semiclassical s”
• Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in R2 (2016, background): “Assumption (f4) is inspired by [18, 52], but here we have the extra difficulty to handle integrals where both the two nonlinearities F (s) and sf(s) appear simultaneously.”