A bounded resonance problem for semilinear elliptic equations (Q2467073)

The authors consider the problem \[ -\Delta u = f(x,u) \text{ in } \Omega,\quad u \in H_0^1(\Omega), \] where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain and \(f:\Omega \times \mathbb{R} \to \mathbb{R}\) is a differentiable function of the form \[ f(x,t) = \lambda_k t + g(x,t), \] where \(0<\lambda_1 < \lambda_2 < \cdots < \lambda_j < \cdots\) are the eigenvalues of \(-\Delta\) on \(H_0^1(\Omega)\), and \(g\) satisfies: \((g_1)\) \(g\) is bounded, \((g_2)\) \(\lim_{|t|\to \infty} g_t'(x,t)=0\), uniformly in \(\Omega\). They also suppose that: \(f(x,0)\equiv 0\); one of the above conditions holds; \((F_0^{\pm})\) \(f_t'(x,0) = \lambda_m\) for some \(m \in \mathbb{N}\), there exists \(r>0\) such that \[ \pm (2F(x,t)-\lambda_m t^2) \geq 0,\quad x \in \Omega,~|t| \leq r. \] As their first results the authors obtain one nonzero solution under the above conditions and also some technical relations between the numbers \(k\) and \(m\). As the second result they obtain three nonzero solutions assuming \((g_1), (g_2)\), \(k>3\) and \(f_t'(x,0)<\lambda_1\). In the last result of the paper it is assumed \((g_1)\), \((g_2)\), \((F_0^+)\) or \((F_0^-)\), \(k>3\), \(m>2\) and the existence of \(t_0 \neq 0 \) such that \(f(x,t_0) \equiv 0\). Under these conditions the authors obtain five nonzero solutions, two of them are with constant sign and the others change sign. For the proofs the authors use Morse Theory and a modification of a penalization scheme due to \textit{A. Masiello} and \textit{L. Pisani} [Ann. Math. Pura Appl., IV. Ser. 171, 1--13 (1996; Zbl 0871.35042)].