Semicoercive variational problems at resonance: An abstract approach. (English) Zbl 0727.35056

The authors consider the Dirichlet problem \[ (*)\quad -\Delta u- \lambda_ 1u+g(x,u)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(\Omega \subset {\mathbb{R}}^ N\) (N\(\geq 1)\) is bounded and \(\lambda_ 1\) is the first eigenvalue of (-\(\Delta\)) on \(H^ 1_ 0(\Omega)\). The Carathéodory function g: \(\Omega\times {\mathbb{R}}\to {\mathbb{R}}\) satisfies \[ | g(x,u)| \leq a| u|^{q-1}+b(x), \] where \(q<\infty\) if \(N=2\), \(q<2N/(N-2)\) if \(N\geq 3\), and \(b(x)\in L^{q'}(\Omega)\), \(1/q+1/q'=1\). In case \(N=1\) it is assumed that for any \(r>0\), \(\sup_{| u| \leq r}| g(x,u)| \in L^ 1(\Omega).\)
The associated functional to problem (*) \[ F(u)=1/2\int_{\Omega}[| \nabla u|^ 2-\lambda_ 1| u|^ 2] dx+\int_{\Omega}\int^{u}_{s=0}g(x,s) ds dx=:A(u)+B(u) \] is a weakly lower semicontinuous \(C^ 1\) functional on \(H^ 1_ 0\), whose critical points are the weak solutions of (*). If F(u) is coercive, i.e. F(u)\(\to \infty\) as \(\| u\| \to \infty\) in \(H^ 1_ 0\), then F has a minimum and consequently (*) has a weak solution. The authors mainly study the coercivity of functionals \(F=A+B\) where A is semicoercive with respect to a subspace and B is coercive on a complementary subspace. Applications are given to the existence of solutions of problem (*).


35J65 Nonlinear boundary value problems for linear elliptic equations
49J20 Existence theories for optimal control problems involving partial differential equations
35J15 Second-order elliptic equations