We study the existence of positive solutions of the quasilinear problem ⎧ $-{\Delta }_{N}u+{V\left(x\right)\left|u\right|}^{N-2}u={f(u,|\nabla u|}^{N-2}\nabla u)$, $x\in {ℝ}^N$, ⎨ ⎩ u(x) > 0, $x\in {ℝ}^N$, where ${\Delta }_{N}u={div\left(\right|\nabla u|}^{N-2}\nabla u)$ is the N-Laplacian operator, $V:{ℝ}^N\to ℝ$ is a continuous potential, $f:ℝ\times {ℝ}^N\to ℝ$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

We study the existence and nonexistence of positive solutions of the nonlinear equation $$-{\Delta }_{p\left(x\right)}u=\lambda k\left(x\right)u^q\pm h\left(x\right)u^r\text{in}\Omega ,u=0\text{on}\partial \Omega$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a regular bounded open domain in ${ℝ}^N$ and the $p\left(x\right)$-Laplacian $${\Delta }_{p\left(x\right)}u:={\text{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$$ is introduced for a continuous function $p\left(x\right)>1$ defined on $\Omega$. The positive parameter $\lambda$ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive solutions...

Consider a class of elliptic equation of the form $$-\Delta u-\frac{\lambda }{{\left|x\right|}^2}u=u^{2^{*}-1}+\mu u^{-q}\text{in}\Omega \setminus \left\{0\right\}$$ with homogeneous Dirichlet boundary conditions, where $0\in \Omega \subset {ℝ}^N$($N\ge 3$), $0<q<1$, $0<\lambda <{(N-2)}^2/4$ and $2^{*}=2N/(N-2)$. We use variational methods to prove that for suitable $\mu$, the problem has at least two positive weak solutions.