This analysis reveals nonexistence of solutions for nonlinear elliptic Dirichlet problems in bounded domains, highlighting implications of domain structure.
In this paper, we give some examples of non-star-shaped bounded domains of R² where, for a class of nonlinear elliptic Dirichlet problems involving supercritical Sobolev exponents, there exists only the trivial identically zero solution (notice that a well-known result of Pohozaev concerns only the star-shaped domains).Unlike the case of previous papers (Molle and Passaseo (2020, 2021, 2023, 2025)) where we proved nonexistence results for nontrivial solutions in thin domains sufficiently close to prescribed curves, in the present paper, the domains do not need to be thin.
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Molle et al. (2025) studied this question.
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