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In this paper, we study the multiplicity and concentration of positive solutions for the following ▫$(p, q)$▫-Laplacian problem: ▫$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb{R}}^{N}, ...

In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: ▫$$\begin{cases}-(\varepsilon^2a+\varepsilon b\int _{\mathbb{R}^3}|\nabla u|^2 dx) \Delta u + V(x)u = f(u)+\gamma u^5 & \text{in} \; \mathbb{R}^3, \\ u \in H^1(\mathbb ...

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian ▫$A_{1/2}$▫ in a smooth bounded domain ▫$\Omega\subset \mathbb{R}^n$▫ (▫$n\geq 2$▫) and with zero Dirichlet boundary conditions. Namely, our simple ...