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January 2017, 16(1): 243-252. doi: 10.3934/cpaa.2017011
Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity
Anouar Bahrouni
Mathematics Department, University of Monastir, Faculty of Sciences, Monastir, 5019, Tunisia
Received April 2016 Revised August 2016 Published November 2016
In this paper we consider the following perturbed nonlocal problem with exponential nonlinearity
$\begin{cases}-\mathcal{L}_{K}u+ \left|u\right|^{p-2}u+h(u)= f \ \ \ \ \ \mbox{in} \ \ \Omega,\\u=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \ \mathbb{R}^{N}\setminus \Omega,\end{cases}$
where $s\in (0, 1)$, $N=ps$, $p\geq 2$ and $f\in L.{\infty}(\mathbb{R}^{N})$. First, we generalize a suitable Trudinger-Moser inequality to a fractional functional space. Then, using the Ekeland's variational principle, we prove the existence of a solution of problem (1).
Keywords: Integrodifferential operators, exponential nonlinearity, TrudingerMoser inequality, existence of solution, variational method.
Mathematics Subject Classification: Primary: 35J60; 35J91; Secondary: 58E30.
Citation: Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
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