We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem $$\left\{\begin{array}{c}{\displaystyle -{\left[M\left({\int }_Q{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{N+ps}}}\mathrm{d}x\mathrm{d}y\right)\right]}^{p-1}{(-\Delta )}_p^{s}u=\lambda h(x,u)\text{in}\Omega ,}\hfill \\ u=0\text{on}{ℝ}^N\setminus \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded smooth domain of ${ℝ}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q={ℝ}^{2N}\setminus (C\Omega \times C\Omega )$, $\lambda >0$ is a numerical parameter, $M$ and $h$ are continuous functions.

This paper discusses the existence and multiplicity of solutions for a class of $p\left(x\right)$-Kirchhoff type problems with Dirichlet boundary data of the following form $$\left\}\begin{array}{cc}-\left(a+b{\int }_{\Omega }\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}dx\right){\mathrm{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)=f(x,u),\hfill & in\Omega \hfill \\ u=0\hfill & on\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a smooth open subset of ${ℝ}^N$ and $p\in C\left(\overline{\Omega }\right)$ with $N<p^{-}={inf}_{x\in \Omega }p\left(x\right)\le p^{+}={sup}_{x\in \Omega }p\left(x\right)<+\infty$, $a$, $b$ are positive constants and $f:\overline{\Omega }\times ℝ\to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x\in {ℝ}^N$, $u\in H¹\left({ℝ}^N\right)$, where λ is a positive parameter, a and b are positive functions, while $f:ℝ\to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations. In order to study the existence of solutions we replace these two problems with their multivalued approximations and, for the first problem, we estabilish an existence result while for the second problem we prove the existence of multiple nontrivial solutions. The approach used is variational.

The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation $$u_{tt}-\alpha u_{xx}-\beta u_{xxtt}=\varphi {\left(u_x\right)}_x$$ are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods $$u_{tt}-\left[a_0+n{a}_1{\left(u_x\right)}^{n-1}\right]u_{xx}-a_2{u}_{xxtt}=0.$$