Nonlinear Hammerstein integral equations via local linking and mountain pass
The author considers nonlinear Hammerstein type integral equations \[ u(x) = \int_{\Omega} k(x,y) f(y,u(y)) dy,\;\;\; x\in \Omega, \tag{1} \] where \(\Omega\) is a bounded closed subset of \({\mathbb R}^n\), \(k(\cdot,\cdot)\) is a continuous symmetric kernel and \(K\) is the corresponding linear integral operator, \(N_f\) is a nonlinear superposition (Nemytsky) operator generated by a continuous function \(f: \Omega\times {\mathbb R} \rightarrow {\mathbb R}\). The solvability of the equation (1) in \(C(\Omega)\) of the equation (1) is reduced to the solvability of the following operator equation in \(L^2(\Omega)\) \[ v = K^{\frac{1}{2}} N_f K^{\frac{1}{2}} v. \tag{2} \] The solvability of the equation ({2}) is studied on the basis of the following notion of local linking. Definition. Let \(W = (W, \|\cdot\|)\) be a real Banach space and let \(W = W_1 \oplus W_2\) be a direct sum decomposition of \(W\) with \(W_1\) being finite dimensional. Assume that the origin is a critical point of a real valued function \(\Phi\) defined on \(W\) (\(\Phi \in C^1(W, {\mathbb{R}})\)) and \(\Phi(0) = 0\). It is said that \(\Phi\) has a local linking near the origin if for some \(r > 0\) we have \[ \Phi(u_1) \leq 0,\quad u_1\in W_1,\quad \|u_1\| \leq r, \] \[ \Phi(u_2) > 0,\quad u_2\in W_2,\quad 0 < \|u_2\| \leq r. \] A number of unique and multiple solvability results for integral equation ({1}) is obtained by using the approach developed by the author earlier on the basis of the above notion.
scientific article; zbMATH DE number 6004391
scientific article