# zbMATH — the first resource for mathematics

Perturbations of quasilinear elliptic equations and Fredholm manifolds. (English. Russian original) Zbl 0625.35029
Math. USSR, Sb. 58, 223-243 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 2(6), 222-242 (1986).
Let us consider a quasi-linear elliptic boundary problem in a bounded domain $$\Omega \subset R^ n$$ with smooth boundary $$\partial \Omega:$$ $(1)\quad -\Delta u(x)+V(x)u(x)-f_{\epsilon}(u,\nabla u,x)=0,$ $(2)\quad 0\not\equiv u(x)\in W^ 2_ p(\Omega)\cap\overset\circ H^ 1(\Omega),\quad V(x)\in W^ 1_ p(\Omega)=B,\quad p>n.$ The function $$f_{\epsilon}(u,\zeta,x)$$ is continuous with respect to x and of class $$C^ k$$, $$k\geq 1$$, with respect to $$\epsilon$$, u, $$\zeta$$. Theorem 1 claims that the set $$\Psi$$ of the pairs (u($$\cdot),V(\cdot))$$ which satisfy (1), (2) with $$\epsilon =0$$ forms a Fredholm manifold of class $$C^ k$$, possessing an additional property which we call “quasi finite-dimensionality”. The projection map $$\pi$$ : $$\psi\to B$$, (u,V)$$\mapsto V$$ is Fredholm, too; ind $$\pi$$ $$=0.$$
Let $$S\subset B$$ be the set of the singular values of $$\pi$$. Theorem 2 is an improved version of the Morse-Smale theorem for the map $$\pi$$, the set S is of first category and $$\mu (S)=0$$ for every Gaussian measure $$\mu$$ on B. If $$V=V_ 0(x)\in (B\setminus S)\cap \pi (\Psi)$$ and $$\epsilon >0$$ is small enough then the problem (1), (2) has solutions u(x), which are at a short distance of the non-empty discrete set $$\pi^{-1}(V_ 0)$$. For some problems it is proved that $$\pi (\Psi)=B.$$
Analogous results are obtained for the following quasi-linear eigenvalue problem: $(3)\quad -\Delta u+V(x)u=\lambda f_{\epsilon}(u,\nabla u,x),\int | \nabla u|^ 2+V(x)u^ 2dx=s,$ where s is a real parameter and pair (u(x),$$\lambda)$$ is a solution. Some results for problem (1), (2) are obtained as corollaries.

##### MSC:
 35J60 Nonlinear elliptic equations 35B20 Perturbations in context of PDEs 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J20 Variational methods for second-order elliptic equations
Full Text: