In this paper we use the topological degree and the fountain theorem to study the existence of weak solutions for a fractional $p$-Laplacian equation in a bounded domain. For the nonlinearity $f$, we consider two situations: (1) the non-resonance case where $f$ is $(p-1)$-asymptotically linear at infinity; (2) the resonance case where $f$ satisfies the Landesman-Lazer type condition.

In our current work we investigate the following critical quasilinear Schrödinger equation

$$-\mathrm{\Delta }\mathrm{\Theta }+\mathcal{V}(x)\mathrm{\Theta }-\mathrm{\Delta }({\mathrm{\Theta }}^2)\mathrm{\Theta }=|\mathrm{\Theta }{|}^{{22}^{\ast }-2}\mathrm{\Theta }+\lambda \mathcal{K}(x)g(\mathrm{\Theta }),x\in {\mathbb{R}}^N,$$

where $N\ge 3$, $\lambda >0$, $\mathcal{V},\mathcal{K}\in C({\mathbb{R}}^N,{\mathbb{R}}^{+})$ and $g\in C(\mathbb{R},\mathbb{R})$ has a quasicritical growth condition. We use the dual approach and the mountain pass theorem to show that the considered problem has a positive solution when $\lambda$ is a large parameter.

In this paper we study a fourth-order differential equation with Riemann-Stieltjes integral boundary conditions. We consider two cases, namely when the nonlinearity satisfies superlinear growth conditions (we use topological degree to obtain an existence theorem on nontrivial solutions), when the nonlinearity satisfies a one-sided Lipschitz condition (we use the method of upper-lower solutions to obtain extremal solutions).