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.

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $-{div\left(\right|x|}^{-ap}{\left|\nabla u\right|}^{p-2}{\nabla u)+h\left(x\right)|u|}^{p-2}u=g\left(x\right){\left|u\right|}^{r-2}u$, x ∈ Ω, ⎨ ⎩ ${\left|x\right|}^{-ap}{\left|\nabla u\right|}^{p-2}\partial u/\partial \nu =\lambda f\left(x\right){\left|u\right|}^{q-2}u$, x ∈ ∂Ω, where Ω is an exterior domain in ${ℝ}^N$, that is, $\Omega ={ℝ}^N\setminus D$, where D is a bounded domain in ${ℝ}^N$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method,...

The existence of solutions for boundary value problems for a nonlinear discrete system involving the $p$-Laplacian is investigated. The approach is based on critical point theory.

We establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧$\Delta \left(\right|\Delta u_i{|}^{p-2}\Delta u_i)=\lambda F_{u_i}(x,u₁,\dots ,uₙ)$ in Ω, ⎨ ⎩$u_i=\Delta u_i=0$ on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.