Existence of positive solutions for a class of biharmonic problems with exponential growth via Nehari manifold method
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In this paper we will be concerned with the problem $$\Delta \left(a(|\Delta u|^p)|\Delta u|^{p-2}\Delta u\right)=f(u)\quad \mbox{in}~~\Omega, \qquad u=\frac{\partial u}{\partial n}=0~~\mbox{on}~~~\partial \Omega,$$ where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain with $1<p<\frac{N}{2},$ $f : \R\rightarrow\R$ is a superlinear continuous function with exponential subcritical or exponential critical growth and the function $a$ is $C^1.$ 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.
Abstract
In this paper we will be concerned with the problem $$\Delta \left(a(|\Delta u|^p)|\Delta u|^{p-2}\Delta u\right)=f(u)\quad \mbox{in}~~\Omega, \qquad u=\frac{\partial u}{\partial n}=0~~\mbox{on}~~~\partial \Omega,$$ where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain with $1<p<\frac{N}{2},$ $f : \R\rightarrow\R$ is a superlinear continuous function with exponential subcritical or exponential critical growth and the function $a$ is $C^1.$ 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.