Multiple solutions of semilinear elliptic equations with one-sided growth conditions (Q5931734)

There is considered the following semilinear elliptic problem: NEWLINE\[NEWLINE \begin{aligned} -\Delta u+ g(x,u) = 0 \quad & \text{in} \quad \Omega, \\ u=0 \quad & \text{on} \quad \partial\Omega, \end{aligned} NEWLINE\]NEWLINE where \(\Omega\) is a open subset of \(\mathbb R^n, n\geq 3,\) the Caratheodory function \(g:\Omega\times \mathbb R\to \mathbb R\) satisfies a ``one-sided'' growth condition like NEWLINE\[NEWLINE s(g)s\geq -a(x)|s|-b|s|^{\frac{2n}{n-2}} NEWLINE\]NEWLINE or NEWLINE\[NEWLINE sg(s)\leq a(x)|s|+b|s|^{\frac{2n}{n-2}}. NEWLINE\]NEWLINE In this case the functional \(f\): NEWLINE\[NEWLINE f(u)=\frac 12 \int\limits_\Omega |Du|^2 dx + \int\limits_\Omega G(x,u) dx, G(x,s):=\int\limits^s_0 g(x,t) dt, NEWLINE\]NEWLINE is no longer of class \(C^1\) on \(W_0^{1,2}(\Omega)\). Therefore some nonsmooth critical point theory is developed and, as consequence, some multiplicity results are obtained.