Opuscula Math. 36, no. 2 (), 253-264
http://dx.doi.org/10.7494/OpMath.2016.36.2.253
Opuscula Mathematica

# Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

Abstract. We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a $$p(x)$$-biharmonic operator $\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}$ where $$\Omega$$ is a smooth bounded domain in $$\mathbb{R}^N$$, $$p\in C(\overline{\Omega})$$, $$\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$$ is the $$p(x)$$-biharmonic operator, and $$\lambda\gt 0$$ is a parameter. We establish sufficient conditions under which there exists a positive number $$\lambda^{*}$$ such that the above problem has at least two nontrivial weak solutions for each $$\lambda\gt\lambda^{*}$$. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces $$L^{p(x)}(\Omega)$$ and $$W^{k,p(x)}(\Omega)$$.
Keywords: critical points, $$p(x)$$-biharmonic operator, weak solutions, mountain pass lemma.
Mathematics Subject Classification: 35J66, 35J40, 35J92, 47J10.