The classical linear duality method in some semilinear and noncoercive Dirichlet problems

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Abstract

We study the following semilinear system:

$(*)\left\{\begin{array}{l} u \in W_{0}^{1,2}(\Omega):-\operatorname{div}(M(x) D u)+u=\theta \psi|\psi|^{p^{\prime}-2}+f(x) ; \\ \psi \in W_{0}^{1,2}(\Omega):-\operatorname{div}(M(x) D \psi)+\psi=u|u|^{p-2} \end{array}\right.$

and we prove the existence of bounded weak solutions in W1,2 0 (Ω). Even if the system is nonlinear, we use a duality method. We dedicate this paper to Patrizia Pucci certain that she will help us for the study of (*) with nonlinear principal part.

Keywords

Dirichlet problems / duality methods / semilinear equations / bounded solutions / elliptic systems

CHANGE HISTORY

A Correction to this paper has been published:
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This article has been retracted.Please see the Retraction Notice for more detail:
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References

[1] H. Brezis, W. A. Strauss, Semi-linear second-order elliptic equations in L¹ , J. Math. Soc. Japan, 25 (1973), 565-590. https://doi.org/10.2969/jmsj/02540565

[2] G. Stampacchia, Le probléme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-257. https://doi.org/10.5802/aif.204