A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth

Abstract

In this paper we give sufficient conditions for the existence of solutions of a biharmonic equation of the form$ \Delta^2 u + V(x)u = f(u)$ in $\mathbb{R}^4$where $V$ is a continuous positive potential bounded away from zero and the nonlinearity $f(s)$ behaves like $e^{\alpha_0 s^2}$ at infinity for some $\alpha_0>0$. In order to overcome the lack of compactness due to the unboundedness of the domain $\mathbb{R}^4$, we require some additional assumptions on $V$. In the case when the potential $V$ is large at infinity we obtain the existence of a nontrivial solution, while requiring the potential $V$ to be spherically symmetric we obtain the existence of a nontrivial radial solution. In both cases, the main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$.