Kirchhoff elliptic problems with asymptotically linear or superlinear nonlinearities

Abstract

We establish the existence and multiplicity of solutions for Kirchhoff elliptic problems of type $$\begin{aligned} -m\left( \mathop \int \limits _{\mathbb {R}^3} |\nabla u|^2 \mathrm{{d}}x\right) \Delta u = f(x,u), \quad x \in \mathbb {R}^3, \end{aligned}$$ where $$m:\mathbb {R}_+\rightarrow \mathbb {R}$$ is continuous, positive and satisfies appropriate growth and/or monotonicity conditions. We consider the cases that f is asymptotically $$3-$$ linear or $$3-$$ superlinear at infinity, in an appropriated sense. By using variational methods, we obtain our results under crossing assumptions of the functions m and f with respect to limit eigenvalues problems. In the model case $$m(t)=a+bt$$ , we also prove a concentration result for some solutions when $$b\rightarrow 0^+$$ .