Sign changing solutions of nonlinear elliptic equations (Q1924505)

The paper is concerned with the nonlinear elliptic problem \[ \text{div} (a_\varepsilon(x)Du)+ g(x,u)=0, \quad u^+\not\equiv 0,\;u^-\not\equiv 0\quad\text{ in }\Omega\qquad\text{and}\quad u=0\text{ on }\partial\Omega, \leqno (P_\varepsilon) \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with \(N\geq 1\), \(a_\varepsilon(x)= (a_\varepsilon^{ij}(x))\) is a positive definite symmetric \(N\times N\) matrix for \(\varepsilon>0\); \(u^+\) and \(u^-\) denote respectively the positive and the negative part of \(u\). The main conditions assumed are roughly summarized as follows: the matrix \(a_\varepsilon(x)\) degenerates as \(\varepsilon\to 0\) in \(k\) disjoint connected subsets \(\Omega_1,\dots, \Omega_k\) (the degeneration subsets for \(a_\varepsilon(x)\)) of \(\Omega\); the function \(g(x,t)\) is superlinear and subcritical, and satisfies appropriate growth conditions at both \(t=0\) and \(t=\pm\infty\). By variational methods, an existence and multiplicity result is obtained that, for \(\varepsilon\) small enough, the problem \(P_\varepsilon\) has at least \(k^2\) distinct solutions \(u_{\varepsilon,r,s}\) \((r,s=1,2,\dots,k)\) with exactly two nodal regions, whose positive and negative parts concentrate near two of the degeneration sets.