Combining the approaches of functionals associated with h-concave functions
and fixed point techniques, we study the existence and uniqueness of a
solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) +
? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R)
denotes the space of all continuous functions on [0,T] equipped with
the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0,
T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued
functions in [0,T]xR.
This chapter covers the notion of hyperelasticity—the concept that stress is derived from a strain—energy function–by invoking an analogy between elastic materials and springs. Alternatively, it can be derived by invoking a work inequality; the notion that work is required to effect a cyclic motion of the material.
In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.
Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3