This paper investigates a high even-order nonclassical differential equation with a spectral parameter. We proved that this equation has a countable system of nontrivial solutions if spectral parameter is negative. We consider two cases, one where the spectral parameter is equal to eigenvalues and one where the spectral parameter is not equal to eigenvalues. In both cases, we proved the existence of regular solutions of boundary value problems for this equation. To do this, we combined the Fourier method and the method of a priori estimates. Moreover, we found some conditions for unsolvability of boundary value problems. In addition, for adjoint problems, we proved that there is no complex eigenvalues.
TWO TOPICS IN DEGENERATE ELLIPTIC EQUATIONS INVOLVING A GRADIENT TERM: EXISTENCE OF SOLUTIONS AND A PRIORI ESTIMATES