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

Lucio Boccardo; Pasquale Imparato; Lucio Boccardo; Pasquale Imparato

doi:10.3934/era.2026003

Electronic Research Archive

2026, Volume 34, Issue 1: 48-54. doi: 10.3934/era.2026003

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

Lucio Boccardo ^,,
Pasquale Imparato

Istituto Lombardo & Sapienza Università di Roma, Italy

Received: 13 October 2025 Revised: 17 December 2025 Accepted: 19 December 2025 Published: 29 December 2025

We study the following semilinear system:
$ (*) \begin{cases} { } u\in{W_0^{1,2}(\Omega)}:\; -{\mathop{{\rm{div}}}}(M(x)\,{D} u) + u = \theta\; \psi|\psi|^{p'-2} +f (x); \\ { } \psi\in{W_0^{1,2}(\Omega)}:\;-{\mathop{{\rm{div}}}}(M(x)\,{D}\psi) + \psi = u|u|^{p-2} \end{cases} $
and we prove the existence of bounded weak solutions in $ {W_0^{1, 2}(\Omega)} $. 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.
- Dirichlet problems,
- duality methods,
- semilinear equations,
- bounded solutions,
- elliptic systems
Citation: Lucio Boccardo, Pasquale Imparato. The classical linear duality method in some semilinear and noncoercive Dirichlet problems[J]. Electronic Research Archive, 2026, 34(1): 48-54. doi: 10.3934/era.2026003

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Abstract

We study the following semilinear system:

$ (*) \begin{cases} { } u\in{W_0^{1,2}(\Omega)}:\; -{\mathop{{\rm{div}}}}(M(x)\,{D} u) + u = \theta\; \psi|\psi|^{p'-2} +f (x); \\ { } \psi\in{W_0^{1,2}(\Omega)}:\;-{\mathop{{\rm{div}}}}(M(x)\,{D}\psi) + \psi = u|u|^{p-2} \end{cases} $

and we prove the existence of bounded weak solutions in $ {W_0^{1, 2}(\Omega)} $. 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.

References

[1]	H. Brezis, W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565–590. https://doi.org/10.2969/jmsj/02540565 doi: 10.2969/jmsj/02540565
[2]	G. Stampacchia, Le problème 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 doi: 10.5802/aif.204

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