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Quasilinear Equations Involving Critical Exponent and Concave Nonlinearity at the Origin

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Abstract

We are interested in quasilinear problems as follows:

$$ \left\{ \begin{array}{ll} -\Delta u -u \Delta (u^2)= -\lambda |u|^{q-2}u+|u|^{22^*-2}u+\mu g(x,u), \quad \mathrm{in}~ \Omega ,\\ u=0,\quad \mathrm{on}~ \partial \Omega , \end{array}\right. $$
(p)

where \(\Omega \subset \mathbb {R}^N \)is a bounded domain with regular boundary \(\partial \Omega , N\ge 3, \lambda , \mu > 0,1<q<4,22^*:=4N/(N-2)\) and g has a subcritical growth and possesses a condition of monotonicity. We prove a regularity result for all weak solutions for a modified problem associated to (p) and, introducing a new type of constraint, we demonstrate a multiplicity result for solutions, including a ground state.

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Acknowledgement

J. C. Oliveira Junior would like to thank very much the Universidade de Brasília for the so pleasant production environment, where all this work was developing. The author also thanks the reviewers for corrections and suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, 70.910-900, Brasília, DF, Brazil

    Giovany M. Figueiredo & R. Ruviaro

  2. Departamento de Matemática, Universidade Federal do Tocantins, 77.824-838, Araguaína, TO, Brazil

    J.C. Oliveira Junior

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  1. Giovany M. Figueiredo
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  2. R. Ruviaro
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Correspondence to J.C. Oliveira Junior.

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Figueiredo, G.M., Ruviaro, R. & Junior, J.O. Quasilinear Equations Involving Critical Exponent and Concave Nonlinearity at the Origin. Milan J. Math. 88, 295–314 (2020). https://doi.org/10.1007/s00032-020-00315-6

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