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|& alpha;)-1$$ {\left(1+{\left & VERBAR;x\right & VERBAR;} circumflex {\alpha}\right)} circumflex {-1} $$, & alpha;& ISIN;(0,4)$$ \alpha \in \left(0,4\right) $$, and (1+|x|& beta;)-1$$ {\left(1+{\left & VERBAR;x\right & VERBAR;} circumflex {\beta}\right)} circumflex {-1} $$, & beta;& ISIN;[& alpha;,+& INFIN;)$$ \beta \in \left[\alpha, +\infty \right) $$, respectively. As an application of this result and by using minimax methods, we establish the existence of solutions for the following class of problems: & UDelta;2u-& UDelta;u+V(x)u=K(x)f(x,u)inDouble-struck capital R4,$$ {\Delta} circumflex 2u-\Delta u+V(x)u equal to K(x)f\left(x,u\right)\kern0.30em \mathrm{in}\kern0.30em {\mathrm{\mathbb{R}}} circumflex 4, $$where the nonlinear term f(x,u)$$ f\left(x,u\right) $$ can have critical exponential growth. Furthermore, when & alpha;& ISIN;(0,2)$$ \alpha \in \left(0,2\right) $$ we prove that the solutions belong to the Sobolev space H2Double-struck capital R4$$ {H} circumflex 2\left({\mathrm{\mathbb{R}}} circumflex 4\right) $$ (bound state solutions).