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| Local Ground-State and Mountain Pass Solutions for a $p$-Kirchhoff Equation with Critical Exponent |
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| KeyWord:Integro-differential equation, p-Kirchhoff equation, critical exponent, mountain-pass solution, local ground-state solution |
| Author Name | Affiliation | | Juan Mayorga-Zambrano | Department of Mathematics, Yachay Tech University, Hda. San Jos\'e s/n y Proyecto Yachay, 100119 Urcuqu\'i, Ecuador | | Henry Cumbal-L\'opez | Department of Mathematics, Yachay Tech University, Hda. San Jos\'e s/n y Proyecto Yachay, 100119 Urcuqu\'i, Ecuador | | Daniel Narv\'aez-Vaca | Facultad de Ciencias, Universidad Central del Ecuador, Av. Universitaria s/n, 170129 Quito, Ecuador | | Jordy Cevallos-Ch\'avez | Sim\'on A. Levin Mathematical, Computational and Modeling Science Center, Arizona State University, 1031 South Palm Walk, Tempe, AZ 85281, USA |
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| Abstract: |
| We study a Kirchhoff-type equation where the diffusion coefficient is non-locally affected, the nonlinear diffusion phenomenon is governed by the $p$-Laplace operator and the population supply presents critical growth. The energy functional associated with the equation is not bounded from below so that there is no global ground-state; however, we prove the existence of a positive local ground-state. We also prove that the equation has a positive solution of mountain pass type. The concentration-compactness principle is a main tool in our approach. |
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