A sign-changing solution for a superlinear Dirichlet problem
The authors study the problem \[ \Delta u + f(u) = 0 \quad \text{in}\;\Omega, \quad u = 0 \quad \text{on}\;\partial \Omega. \tag{P} \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\), \(f\in C^1\) is superlinear and subcritical and \(f'(0)\) is stricly less than the first eigenvalue of the Laplacian on \(\Omega\). Under these assumptions the existence of three different solutions of (P) is proved: \(u_1\) (positive), \(u_2\) (negative), and \(u_3\) which changes sign exactly once. For the proof, the authors consider the corresponding functional \(J(u) = \int_{\Omega} (1/2) | \nabla u | ^2 - F(u) \,dx\) on the space \(H^1_0(\Omega)\). The sign-changing solution is then obtained by minimizing \(J\) on the subset \[ S_1 = \{ u \neq 0 : (\nabla J(u), u) = 0, \;u_{+}, u_{-} \neq 0, \;(\nabla J(u_{+}), u_{+}) = 0 \}. \] Their proof also shows the inequality \( J(u_3) \geq J(u_1) + J(u_2) \).
critical level of sign-changing solution
scientific article; zbMATH DE number 1206684
scientific article