an:05251331
Zbl 1147.47049
Anello, Giovanni; Cordaro, Giuseppe
Nonlinear equations involving nonpositive definite linear operators via variational methods
EN
J. Integral Equations Appl. 19, No. 1, 1-12 (2007).
00216948
2007
j
47J05 49J40 47J30 47H30 49J45 49J50
variational method; nonlinear integral equation; Fredholm equation; Hammerstein equation
The paper is concerned with the application of variational methods, namely the variational method due to \textit{B. Ricceri} [J. Comput. Appl. Math. 113, No. 1--2, 401--410 (2000; Zbl 0946.49001)] in showing the existence of solutions for nonlinear equations of the type \(u=K\mathbf{f}(u)\), where \(K:L^{q_{0}}( \Omega) \rightarrow L^{p_{0}}( \Omega) \) is a completely continuous linear operator, \(\mathbf{f}(u)= f( \cdot ,u( \cdot))\) denotes the superposition operator associated with \(f\) such that \(\mathbf{f}(u)\in L^{q}( \Omega ) \) for every \(u\in L^{p}( \Omega) \) and \(2<p<p_{0}\) , \(\frac{1}{p}+\frac{1}{q}=1\), \(\frac{1}{p_{0}}+\frac{1}{q_{0}}=1\).
The problems considered in this research were usually tackled by fixed point methods. The authors of the present study apply variational methods which improve the few known results obtained by critical point theory and which apply the action functional associated with the considered problem fails to be coercive. The authors, contrary to some other results in the field, do not assume \(K\) to be positive definite but require that it has a finite number of negative eigenvalues and put some weak assumption on the nonlinear term which provides that the action functional is not coercive. The results of the paper are applied to solving a Hammerstein integral equation and a nonresonant nonlinear Fredholm integral equation.
Marek Galewski (????d??)
Zbl 0946.49001