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Stable solutions and bifurcation problem for asymptotically linear Helmholtz equations. (English) Zbl 1353.35130

Summary: In this note, we investigate the existence of positive solutions for the Helmholtz equation \(-\Delta u + cu = \lambda f (u)\) on a bounded smooth domain of \(\mathbb{R}^n\) with Dirichlet boundary conditions. Here \(\lambda > 0\), \(c > 0\) are positive constants and \(f\) is a positive nondecreasing convex function, asymptotically linear that is \(\lim_{t\to\infty} \frac{f (t)}{t} = a < \infty\). We show that there exists an extremal parameter \(\lambda^\ast> 0\) but the extremal solution exists and it is regular provided that \(\lim_{t\to\infty}f (t) - at = l < 0\).

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B09 Positive solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
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