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In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations \[ -\Delta_{\gamma} u + a(z)u = f(u) \quad \mbox{in}\ \mathbb{R}^{N}, \] where $\Delta _{\gamma }$ is known as the Grushin operator, $z:=(x,y)\in \mathbb {R}^{m}\times \mathbb {R}^{k}$ and $m+k=N\geqslant 3$ , $f$ and $a$ are continuous function satisfying some technical conditions. In order to overcome some difficulties involving this type of operator, we have proved some compactness results that are crucial in the proof of our main results. For the case $a=1$ , we have showed a Berestycki–Lions type result.

Abstract. In this paper we show existence of positive solutions for a class of quasi-linear problems with Neumann boundary conditions defined in a half-space and involving the critical exponent.

In this paper we show the existence of solution for the following class of semipositone problem P$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$ where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.

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