On the Existence of Three Nontrivial Solutionsof a Resonance Elliptic Boundary Value Problemwith a Discontinuous Nonlinearity
2020
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.
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