Variational method with application to convolution equations
The author considers the problem of solvability for the nonlinear integral equation \(k* u^ 2= | u|\). Here \(u\in L^ 3(\mathbb{R})\), and \(k\in L^{3/2}(\mathbb{R})\) satisfies the conditions: \(k(x)\geq 0\), \(k(x)= k(- x)\) and \(k\) is symmetrically decreasing. The investigation is based on some variational method and results due to \textit{A. Szulkin} [Ann. Inst. Henri Poincaré Anal. Non Linéaire 5, No. 2, 119-139 (1988; Zbl 0661.58009)].
scientific article; zbMATH DE number 739389