American Institute of Mathematical Sciences

$ \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 $f$.

$-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=g(x,u) $

in $\mathbb{R}^N$, $N\geq 3$ and $g(x,u)$ is a superlinear but subcritical function. Applying variational methods we show the existence and multiplicity of solutions provided $\varepsilon$ is sufficiently small enough.

$ -\mathcal{M}_{\lambda,\Lambda}^+ (D^2u)=f(u) $ in $\Omega$,

$ u=0 $ on $\partial \Omega$

and a more general fully nonlinear elliptic equation

$ F(D^2u)=f(u) $ in $\Omega$,

$ u=0 $ on $\partial \Omega$,

where $\Omega$ is a bounded domain in $\mathbb{R}^N, N\geq 3$, $f$ is a locally Lipschitz continuous function with superlinear growth at infinity. We will show that the equation has at least two positive solutions under some assumptions.