Nonlinear Robin problems with a reaction of arbitrary growth
The Robin problem in question is the nonlinear elliptic boundary value problem \(-\operatorname{div} a(\nabla u)=f(x,u)\) in \(\Omega\), \(\partial u/\partial n_a=-\beta(x)|u|^{p-2}\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with a \(C^2\) boundary \(\partial\Omega\), \(a:\mathbb R^N\to\mathbb R^N\) is a continuous strictly monotone function satisfying a set of regularity and growth conditions, and \(\partial u/\partial n_a=(a(\nabla u),n)_{\mathbb R^N}\) with the outward unit normal \(n(x)\) at \(x\in\partial\Omega\). The boundary value function \(\beta\) is a non-negative \(\alpha\)-Hölder continuous function on \(\partial\Omega\) with \(0<\alpha<1\). The reaction \(f\) is a Carathéodory function which, instead of being assumed to satisfy any global growth condition, admits \(x\)-dependent zeros of constant sign. Examples of admissible functions \(a\) include \(a(\xi)=|\xi|^{p-2}\xi\) with \(1<p<\infty\) for the well-known \(p\)-Laplacian, \(a(\xi)=|\xi|^{p-2}\xi+|\xi|^{q-2}\xi\) with \(1<q<p<\infty\), \(a(\xi)=(1+|\xi|^2)^{(p-2)/2}\xi\) with \(1<p<\infty\) for the generalized \(p\)-mean curvature operator, or \(a(\xi)=|\xi|^{p-2}\xi[1+(1+|\xi|^p)^{-1}]\) with \(1<p<\infty\) for an operator arising in plasticity theory. The authors prove multiplicity theorems for the solutions of the Robin problem. Using variational methods based on critical point theory, suitable truncation and perturbation techniques and the Morse theory, they show that the problem has at least three nontrivial solutions, two of them having constant opposite sign and the third one being nodal.
scientific article; zbMATH DE number 6614272
scientific article