EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF BIHARMONIC PROBLEMS WITH EXPONENTIAL GROWTH VIA NEHARI MANIFOLD METHOD

EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF BIHARMONIC PROBLEMS WITH EXPONENTIAL GROWTH VIA NEHARI MANIFOLD METHOD

In this paper we will be concerned with the problem ${\Delta}\,(a{\mid}{\Delta}u{\mid}^p){\mid}{\Delta}u{\mid}^{p-2}{\Delta}u)\;=\,f(u)\;in\;{\Omega},\;u\;=\;{\frac{{\partial}u}{{\partial}n}}\;=\;0\;on\;{\partial}{\Omega},$ where &#x2126; &#x2282; &#x211D;^N is a bounded smooth domain with ${1\,<\,p\,<\,{\frac{N}{2}}}$, f : &#x211D; &#x2192; &#x211D; is a superlinear continuous function with exponential subcritical or exponential critical growth and the function a is C¹. We use as our main tool the Nehari manifold method, supplemented by the application of the quantitative deformation lemma and degree theory results. Our results encompass a broad class of problems.