EXISTENCE OF NONTRIVIAL SOLUTIONS TO POLYHARMONIC EQUATIONS WITH SUBCRITICAL AND CRITICAL EXPONENTIAL GROWTH

By Nguyen Huong Lam, G. Lu. 2012

49 citations recorded by the source corpus.

The main purpose of this paper is to establish the existence of nontrivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adams inequality [1] of Moser-Trudinger type. More precisely, we consider the semilinear elliptic equation \[ \left( -\Delta\right) ^{m}u=f(x,u), \] subject to the Dirichlet boundary condition $u=\nabla u=...=\nabla^{m-1}u=0$, on the bounded domains $\Omega\subset \mathbb{R}^{2m}$ when the nonlinear term $f$ satisfies exponential growth condition. We will study the above problem both in the case when $f$ satisfies the well-known Ambrosetti-Rabinowitz condition and in the case without the Ambrosetti-Rabinowitz condition. This is one of a series of works by the authors on nonlinear equations of Laplacian in $\mathbb{R}^2$ and $N-$Laplacian in $\mathbb{R}^N$ when the nonlinear term has the exponential growth and with a possible lack of the Ambrosetti-Rabinowitz condition (see [23], [24]).

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1e27b20acd59934ced73750fff67319f2b852537

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In Applied Mathematics

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

In Computational Theory and Mathematics

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

Citations in the corpus

• Sharp singular Adams inequalities in high order Sobolev spaces (2011, unknown): “We refer the interested reader to [13], [16], [3], [7], [6], [9], [11], [8], [20], [21], [19], [27, 28], [49], [50] and the references therein.”
• Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications ✩ (2012, unknown): “Existence of solutions to polyharmonic operators and N -Laplacian equations of exponential growth without satisfying the standard Ambrosetti–Rabinowitz condition have been studied by the authors in [41,44,45].”
• Elliptic Equations and Systems with Subcritical and Critical Exponential Growth Without the Ambrosetti–Rabinowitz Condition (2014, background): “Existence of nontrivial and nonnegative solutions to polyharmonic equations of exponential growth and without the AR condition has also been carried out in [23].”
• N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition (2013, unknown): “In the Euclidean spaces, see for example, [1], [2], [3], [8], [5], [7], [18], [19], [15], [16], [32], [31], [25], [26], [27], [28], [39], [40], etc.”
• Adams type inequalities and related elliptic partial differential equations in dimension four (2011, background): “0], CarlesonChang [12], Adimurthi et al. [3]-[7], de Figueiredo-Miyagaki-Ruf [14], Panda [28], J. M. doO´ [16], de Figueiredo-doO-Ruf [13], Silva-Soares [33], Yang-Zhao [38], do´ O-Yang [19], Lam-Lu´ [23] and the references therein. Email address: yunyanyang@ruc.edu.cn(Yunyan Yang) Preprint submitte”