Variational elliptic problems at double resonance (Q1907731)

The problem of the existence of weak solutions \(u\in H^1_0(\Omega)\) for the system \[ - \Delta u= f(x, u)\quad\text{in} \quad \Omega,\quad u= 0\quad\text{on} \partial\Omega\tag{1} \] is studied under the following assumptions: \(|f(x, t)|\leq a|t|^\sigma+ b\), a.e. \(x\in \Omega\) for some \(\sigma\) and \(\lambda_\ell\leq L(x)\), \(K(x)\leq \lambda_{\ell+ 1}\) a.e. \(x\in \Omega\), where \(\lambda_\ell\) is the \(\ell\)th eigenvalue of \(-\Delta\) and \[ L(x)= \liminf_{|t|\to \infty} {2F(x, t)\over t^2},\quad K(x)= \limsup_{|t|\to \infty} {2F(x, t)\over t^2}, \] where \(F(x, t)= \int^t_0 f(x, \tau) d\tau\). Moreover, the function \(H(x, t)= 2F(x, t)- tf(x, t)\) is assumed to satisfy \[ H(x, t)\to - \infty\quad\text{a.e. } x\in \Omega\quad\text{as } |t|\to \infty,\quad H(x, t)\leq M(x)\quad\text{a.e. }x\in \Omega \] for some \(M\in L^1(\Omega)\). Then, under these assumptions and using a variational approach, it is shown that (1) has a weak solution if \(2F(x, t)\geq (\lambda_\ell- \varepsilon) t^2- A_\varepsilon(x)\), \(t\in \mathbb{R}\), a.e. \(x\in \Omega\), \(\varepsilon> 0\) and \(A_\varepsilon\in L^1\) and \(\sup_{v\in V} J(v)< \infty\), where \(J(u)= {1\over 2} \int_\Omega |\nabla u|^2- \int_\Omega F(x, u)\).