Existence and multiplicity of solutions to equations of $N-$Laplacian type with critical exponential growth in $\mathbb{R}^{N}$

By Nguyen Huong Lam, G. Lu. 2011

161 citations recorded by the source corpus.

In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the Ambrosetti-Rabinowitz condition. In spite of a possible failure of the Palais-Smale compactness condition, in this article we apply minimax method to obtain the weak solution to such an equation. In particular, in the case of $N-$Laplacian, using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions. Finally, we will prove the above results when our nonlinearity doesn't satisfy the well-known Ambrosetti-Rabinowitz condition and thus derive the existence and multiplicity of solutions for a much wider class of nonlinear terms $f$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/37c07b80e75f0eaffeb0c05da84d5a6eb113b3ee

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In Applied Mathematics

1. Nguyen Huong Lam
2. G. Lu
3. Caifeng Zhang
4. Hanli Tang
5. Y. Yang
6. Federica Sani
7. Jiuyi Zhu
8. Lu Zhang

In Computational Theory and Mathematics

1. Nguyen Huong Lam
2. G. Lu
3. Caifeng Zhang
4. Hanli Tang
5. Y. Yang
6. Federica Sani
7. Jiuyi Zhu
8. Lu Zhang

Citations in the corpus

• Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs (2014, unknown): “Moreover, our restriction on the norm ∥∇Hu∥LQ (H) ≤ 1 of the function u is much weaker than ∥∇Hu∥LQ (H) + ∥u∥ LQ (H) ≤ 1 which was assumed in Lam and Lu (2012) [16].”
• EXISTENCE OF NONTRIVIAL SOLUTIONS TO POLYHARMONIC EQUATIONS WITH SUBCRITICAL AND CRITICAL EXPONENTIAL GROWTH (2012, background): “This paper, along with a series of works by the authors [23], [24], is an attempt to study the existence of nontrivial nonnegative solutions to nonlinear equations when the nonlinear term satisfies the subcritical and critical exponential growth but does not satisfy the well-known Ambrosetti-Rabinow”
• Sharp singular Adams inequalities in high order Sobolev spaces (2011, background): “re the nonlinear term behaves like eα|u| n n−m as |u| → ∞. There has been a vast literature in this direction. We refer the interested reader to [10], [13], [3], [4], [6], [8], [5], [15], [16], [14], [21, 22] and the references therein. In this paper, we will first establish a sharp inequality of exp”
• Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition (2014, unknown): “by the authors in [21, 22].”
• Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications ✩ (2012, background): “Now, by standard arguments (see [24,40]), we can deduce a contradiction.”