On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity

• Published in: Differential equations 2020-07, Vol.56 (7), p.831-841
• Main Authors: Pavlenko, V. N., Potapov, D. K.
• Format: Article
• Language: English
• Subjects: Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266120070034

We study the homogeneous Dirichlet problem for a second-order elliptic equation with a nonlinearity discontinuous in the state variable in the resonance case. A class of resonance problems that does not overlap with the previously investigated class of strongly resonance problems is singled out. Using the variational method, we establish a theorem on the existence of at least three nontrivial solutions of the problem under study (the zero is its solution). In this case, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on the discontinuities of the nonlinearity only on a set of measure zero. We give an example of a nonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition is obtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest. Applications of the theorem to problems with a parameter are considered. The existence of nontrivial (including semiregular) solutions of the problem with a parameter for an elliptic equation with a discontinuous nonlinearity for all positive values of the parameter is established.