Existence, Decay, and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Delay Term

This article deals with the study of the higher-order Kirchhoff-type equation with delay term in a bounded domain with initial boundary conditions, where firstly, we prove the global existence result of the solution. Then, we discuss the decay of solutions by using Nakao’s technique and denote polynomially and exponentially. Furthermore, the blow-up result is established for negative initial energy under appropriate conditions.

A multiplicity result for a (p, q)-Schrödinger–Kirchhoff type equation

In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

The Existence of Nontrivial Solution for a Class of Kirchhoff-Type Equation of General Convolution Nonlinearity without Any Growth Conditions

In this paper, we consider the following Kirchhoff-type equation:{u∈H1(RN),−(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα*F(u))f(u)+λg(u),inRN, where a>0, b≥0, λ>0, α∈(N−2,N), N≥3, V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N−2,N). Under certain assumptions on V(x), f(u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods.