We consider a family of elliptic boundary value problems of the type $$-\sum_{i,j=1}^n a_{ij}(x,u)\frac{\partial^2u}{\partial x_i\partial x_j}= f(x,u,\lambda)\quad\hbox{in }\Omega,$$ where $\Omega$ is a bounded domain of ${\bf R}^n$. We prove an existence result for unbounded subcontinua of positive solutions containing $(0,0)$, where $f$ is allowed to be highly discontinuous in $u$. Actually, a function $f$ satifying the assumptions of our main result can be discontinuous, with respect to the second variable, even at all points of its domain $[0,+\infty[$. Some special cases and also examples are presented. Our results improve some existing results in the literature, where the set of discontinuity points must have null Lebesgue measure.
On the structure of the solution set of elliptic eigenvalue boundary-value problems with highly discontinuous right-hand side
F. Cammaroto;P. Cubiotti
;
2025-01-01
Abstract
We consider a family of elliptic boundary value problems of the type $$-\sum_{i,j=1}^n a_{ij}(x,u)\frac{\partial^2u}{\partial x_i\partial x_j}= f(x,u,\lambda)\quad\hbox{in }\Omega,$$ where $\Omega$ is a bounded domain of ${\bf R}^n$. We prove an existence result for unbounded subcontinua of positive solutions containing $(0,0)$, where $f$ is allowed to be highly discontinuous in $u$. Actually, a function $f$ satifying the assumptions of our main result can be discontinuous, with respect to the second variable, even at all points of its domain $[0,+\infty[$. Some special cases and also examples are presented. Our results improve some existing results in the literature, where the set of discontinuity points must have null Lebesgue measure.Pubblicazioni consigliate
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