Existence of solutions for \(p\)-Laplacian type equations. (English) Zbl 0858.35044
The authors study the nonlinear boundary value problem
\[
\text{div }A(-\nabla u)=f(x,u,\nabla u)\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega,\tag{\(*\)}
\]
where \(\text{div }A(-\nabla u)\) is a nonlinear operator of \(p\)-Laplacian type. They prove existence of a nontrivial solution of this problem in the following cases:
\[
(1)\quad f(x,u,\nabla u)=h-u|\nabla u|^{p-2}, \qquad (2)\quad f(x,u,\nabla u)=g(x,u).
\]
Here, \(h\in L^\infty\) and \(g\) is superlinear and subcritical in \(u\). In the first case, an approximation procedure is used, where \(u|\nabla u|^{p-2}\) is replaced by \(u|\nabla u|^{p-2}/(1+ \varepsilon|u|\;|\nabla u|^{p-2})\). The corresponding solution, \(u_\varepsilon\), is shown to converge to a solution of \((*)\) as \(\varepsilon\to0\), For case (2), the mountain pass lemma is used to construct a pair of nontrivial solutions \(u_-\leq 0\leq u_+\).
Reviewer: K.Pflüger (Berlin)
MSC:
35J65 | Nonlinear boundary value problems for linear elliptic equations |
35J20 | Variational methods for second-order elliptic equations |
35A35 | Theoretical approximation in context of PDEs |