## 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 (*).

### MSC:

 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

### Keywords:

semilinear Dirichlet problem; coercivity; existence