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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
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