On a weighted Adams type inequality and an application to a biharmonic equation.

This paper deals with an improvement of a class of Adams type inequalities involving potentials V$$ V $$ and weights K$$ K $$, which can decay to zero at infinity as (1+|x|α)−1$$ {\left(1+{\left|x\right|}^{\alpha}\right)}^{-1} $$, α∈(0,4)$$ \alpha \in \left(0,4\right) $$, and (1+|x|β)−1$$ {\left(1+{\left|x\right|}^{\beta}\right)}^{-1} $$, β∈[α,+∞)$$ \beta \in \left[\alpha, +\infty \right) $$, respectively. As an application of this result and by using minimax methods, we establish the existence of solutions for the following class of problems: Δ2u−Δu+V(x)u=K(x)f(x,u)inℝ4,$$ {\Delta}^2u-\Delta u+V(x)u=K(x)f\left(x,u\right)\kern0.30em \mathrm{in}\kern0.30em {\mathrm{\mathbb{R}}}^4, $$where the nonlinear term f(x,u)$$ f\left(x,u\right) $$ can have critical exponential growth. Furthermore, when α∈(0,2)$$ \alpha \in \left(0,2\right) $$ we prove that the solutions belong to the Sobolev space H2ℝ4$$ {H}^2\left({\mathrm{\mathbb{R}}}^4\right) $$ (bound state solutions). [ABSTRACT FROM AUTHOR]