On the solvability of a quasilinear elliptic resonance problem near its first eigenvalue
Let \(\Omega\subset\mathbb{R}^N\), \(N\geq 1\), be a bounded smooth domain and \(\alpha\) an ordered \(N\)-tuple of nonnegative integers. For \(u,v\in W_0^{m,p} (\Omega)\), \(m\geq 1\), \(1<p <\infty\), define the operator \(Q\) by \(Q(u)(v)= \sum_{ |\alpha|\leq m}\int_\Omega A_\alpha D^\alpha v dx\), where the coefficients \(A_\alpha\) are Caratheodory functions which may depend on \(x\) and the derivatives of \(u\) and satisfy appropriate growth conditions. Using Galerkin-type techniques and Brouwer's fixed point theorem, the author studies elliptic resonance problems. Given a Carathéodory function \(f\) and \(G\in(W_0^{m,p}(\Omega))^*\), he proves the existence of a function \(u\in W_0^{m, p}(\Omega)\) such that \[ Q(u)(v) -\mu\int_\Omega (\text{sgn} u)|u|^{p-1}v dx+ \int_\Omega f(x,u) v dx= G(v) \] for all \(v\in W_0^{m,p} (\Omega)\), \(\mu\) being the first eigenvalue of \(Q\). The purpose of this article is to extend previous results to the case that \(f\) has no growth restriction as \(|u|\to\infty\) and in which the Landesman-Lazer condition is not satisfied.
scientific article; zbMATH DE number 1309048