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Nonlocal elliptic equation with critical exponential growth and resonance in high-order eigenvalues. (English) Zbl 1484.35214

Summary: In this paper we are interested in the following nonlocal elliptic equation \[ \begin{cases} -\Delta u=\lambda_k u+\bigg[ \dfrac{1}{|x|^{\mu}}\ast G(x,u)\bigg] g(x,u) & \text{in } \Omega, \\ u=0 & \text{on } \partial\Omega, \end{cases} \] where an open \(\Omega\subset\mathbb{R}^2\) is bounded with smooth boundary. The nonlinearity \(g(x,s)\) has the critical exponential growth in the sense of the Trudinger-Moser inequality and \(\lambda_k\) denotes the \(k\)th eigenvalue of \((-\Delta,H_0^1 (\Omega))\), \(k\geq 2\). Employing variational methods we prove the existence of a nontrivial solution for this nonlocal elliptic problem.

MSC:

35J61 Semilinear elliptic equations
35B33 Critical exponents in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
Full Text: DOI

References:

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