Resonance problems for the \(p\)-Laplacian with a nonlinear boundary condition
Consider the following problem \[ \begin{cases} & \Delta_p u =| u| ^{p-2}u-g_1(u)+h_1(x),\;\;\text{in}\;\Omega\\ & | \nabla u| ^{p-2}\frac{\partial u}{\partial \nu} =\lambda | u| ^{p-2}u+g_2(u)+h_2(x),\;\;\text{on}\;\partial \Omega \end{cases}\eqno(1) \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary, \(-\Delta_p u=-\text{div}(| \nabla u| ^{p-2}\nabla u)\) is the \(p\)-Laplacian with \(p>1\), the perturbation \(g_1,g_2 \in C(\mathbb R,\mathbb R), h_1(x)\in L^{p'}(\Omega), h_2(x)\in L^{p'}(\partial \Omega)\) \((p'=p/(p-1))\). Under the Landesman-Lazer type conditions, using a variational method, the authors prove the existence of weak solutions for (1) in \(W^{1,p}(\Omega)\). This paper is a continuation of the article by \textit{P. Drábek} and \textit{S. B. Robinson} [J. Funct. Anal. 169, No. 1, 189--200 (1999; Zbl 0940.35087)], and of one by \textit{S. Martinez} and \textit{J. D. Rossi} [ Electron. J. Differ. Equ. 2003, Paper No. 27, 14 p., electronic only (2003; Zbl 1033.35078)].
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