Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

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

In this paper, we study the problem \[ \left\{\begin{array}{@{}l} -\Delta u + u + V(x)u = u^p,\\[3pt] -\Delta V = \lambda u^2, \quad \displaystyle\lim_{|x| \to +\infty} V(x)=0, \end{array}\right. \] where u, V : ℝ3 → ℝ are radial functions, λ > 0 and 1 < p < 5. We give multiplicity results, depending on p and on the parameter λ.

Multiple bound states for the Schroedinger-Poisson problem

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.

/pdf/a-guide-to-the-choquard-equation-4ufiqlw479.pdf

A guide to the Choquard equation

This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problem − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = λ f ( x , u ) in Ω and u = 0 on ∂ Ω , where Ω is a smooth bounded domain of R N , a , b > 0 , λ > 0 and f : Ω × R → R is a Caratheodory function satisfying some further conditions. We obtain a sequence of a.e. positive weak solutions to the above problem tending to zero in L ∞ ( Ω ) with f being more general than that of [K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006) 246–255; Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463].

Infinitely many positive solutions for Kirchhoff-type problems

Preliminaries.- Functionals Bounded Below.- Even Functionals.- Linking and Homoclinic Type Solutions.- Double Linking Theorems.- Superlinear Problems.- Systems with Hamiltonian Potentials.- Linking and Elliptic Systems.- Sign-Changing Solutions.- Cohomology Groups.

Critical point theory and its applications

In this paper we establish some variant fountain theorems without (P.S.)-type assumption. The abstract results will be used to study the symmetric nonlinear Schrodinger equations and Dirichlet boundary value problems. Under no Ambrosetti–Rabinowitz's superquadraticity condition, we obtain infinitely many large energy and small negative energy solutions respectively.

Wenming Zou

Papers

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

Infinitely many positive solutions for Kirchhoff-type problems

Critical point theory and its applications

Variant fountain theorems and their applications

Homoclinic Orbits for Asymptotically Linear Hamiltonian Systems