Existence and the number of solutions of nonresonant semilinear equations and applications to boundary value problems (Q5931735)

Using degree theory the author states some surjectivity results for A-proper mappings. These results are then applied to what the author calls semiabstract nonresonance problems NEWLINE\[NEWLINEAu+g(x,u,Du,\dots,D^{2m- 1}u)u+f(x,Du,\dotsc,D^{2m}u)=h(x)NEWLINE\]NEWLINE in \(V\) where \(Q\subset\mathbb{R}^n\) is a bounded domain, \(V\) is a closed subset of \(W_2^{2m}(Q)\) containing the test functions, \(L:V\to L_2(Q)\) is a linear map with closed range and \(L_1\) is the restriction of \(L\) to a closed subspace \(V_1\) of \(V\) such that each eigenvalue of \(L_1\) has finite multiplicity and the corresponding eigenfunctions form a complete set in \(V_1\). Let \(A=A_1+L\) with a linear map \(A_1:V\to L_2(Q)\) and assume that there is a \(\lambda\) which is different from the eigenvalues of \(L_1\) such that \(-A+(\lambda_j-\lambda)I\) is bijective for some eigenvalue \(\lambda_j\) of \(L_1\). The author derives conditions on the existence of solutions \(u\in V\) if \(h\in L_2(Q)\). This result is then applied to the strong solvability of elliptic boundary value problems, to the existence of generalized periodic solutions for nonlinear parabolic and hyperbolic equations always under nonuniform nonresonance conditions. As concrete examples, the author deals with the heat equation, the telegraph equation and the damped beam equation.