In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:

$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$

$M:{\open R}_0^ + \to {\open R}^ + $

In this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problems

where λ ∈ (0, ∞), f is a continuous function with exponential critical growth and V : ℝ2→ ℝ is a continuous function verifying some hypotheses.

In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundary-value problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli–Kohn–Nirenberg exponent.

We establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem −Lpu + |u|p−2u = h(u) in Ωλ, u = 0 on ∂Ωλ, where Ωλ = λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter, Lpu ≐ Δpu + Δp(u2)u and the nonlinear term h(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

We study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson system

in ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

In this paper we study the multiplicity of solutions of the quasilinear elliptic system

\begin{equation} \left. \begin{aligned} -\Delta_pu\amp=\lambda F_u(x,u,v)\amp\amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)\amp\amp\text{in }\varOmega, \\ u=v\amp=0\amp\amp\text{on }\partial\varOmega, \end{aligned} \right\} \end{equation} \tag{S$_\lambda$}

where $\varOmega$ is a strip-like domain and $\lambda>0$ is a parameter. Under some growth conditions on $F$, we guarantee the existence of an open interval $\varLambda\subset(0,\infty)$ such that for every $\lambda\in\varLambda$, the system (S$_\lambda$) has at least two distinct, non-trivial solutions. The proof is based on an abstract critical-point result of Ricceri and on the principle of symmetric criticality.

In this paper, a dual version of the Mountain Pass Theorem and the Generalized Mountain Pass Theorem are extended to functions that are locally Lipschitz only. An application involving elliptic hemivariational inequalities is next examined.