Nontrivial solutions of Kirchhoff-type problems via the Yang index

Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3

Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations 

$$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$

and some of its variants and extensions.

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A guide to the Choquard equation

The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem; in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4-superlinear Kirchhoff type problems without the P.S. condition, one concerning the existence of a nontrivial solution and the other one concerning the existence of sign-changing solutions and multiple solutions, { − ( a + b ∫ Ω | ∇ u | 2 ) △ u = f ( x , u ) in  Ω , u = 0 on  ∂ Ω .

Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition

This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type -M(@!"@W|@?u|^2dx)@Du=f(x,u),[email protected],u=0,[email protected][email protected], where @W is a smooth bounded domain of @?^N, M is a positive function, and f has subcritical growth.

Positive solutions for a quasilinear elliptic equation of Kirchhoff type

On systems of elliptic equations involving subcritical or critical Sobolev exponents

sic) Abstract. This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the Kirchhoff type

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On a class of nonlocal elliptic problems with critical growth

Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity

This paper is concerned with the existence of positive solutions for the following class of nonlocal problem M ( ∫ R N | ∇ u | 2 d x + ∫ R N V ( x ) | u | 2 d x ) [ − Δ u + V ( x ) u ] = λ f ( u ) + γ u τ in R N , where τ = 5 for N = 3 and τ ∈ ( 1 , + ∞ ) for N = 1 , 2 , λ is a positive parameter and γ ∈ { 0 , 1 } . Moreover, M , V , and f are continuous functions satisfying some conditions.

Claudianor O. Alves

Papers

Positive solutions for a quasilinear elliptic equation of Kirchhoff type

On systems of elliptic equations involving subcritical or critical Sobolev exponents

On a class of nonlocal elliptic problems with critical growth

Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity

Nonlinear perturbations of a periodic Kirchhoff equation in RN