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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.

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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