# Elliptic variational problems with mixed nonlinearities

## TL;DR

This paper investigates the existence and multiplicity of positive solutions for a class of elliptic equations with mixed nonlinearities in Euclidean space, extending understanding of solution behavior under various parameter and function conditions.

## Contribution

It provides new results on positive solutions for elliptic equations with combined nonlinearities, considering different dimensional and parameter regimes.

## Key findings

- Existence of positive solutions under certain conditions.
- Multiple solutions are established for specific parameter ranges.
- Results depend on the interplay between nonlinear terms and the domain dimension.

## Abstract

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min\{p^*,q\}$ if $p<N$ while $p<r<q$ if $p\geq N$, and $\lambda>0$ is a parameter.