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Sign changing solutions with clustered layers near the origin for singularly perturbed semilinear elliptic problems on a ball. (English) Zbl 1175.35061

Summary: We study sign changing solutions to equations of the form
\[ -\varepsilon^2\Delta u+u= f(u)\quad\text{in }B, \qquad \partial_\nu u=0\quad\text{on }\partial B, \]
where \(B\) is the unit ball in \(\mathbb R^N\) \((N\geq 2)\), \(\varepsilon\) is a positive constant and \(f(u)\) behaves like \(|u|^{p-1}u\) (but not necessarily odd) with \(1<p<(N+2)/(N-2)\) if \(N\geq 3\), and \(1<p<\infty\) if \(N=2\). We show that for any given positive integer \(n\), this problem has a sign changing radial solution \(v_\varepsilon(|x|)\) which changes sign at exactly \(n\) spheres \(\bigcup_{j=1}^n \{|x|=\rho_j^\varepsilon\}\), where \(0< \rho_1^\varepsilon<\cdots< \rho_n^\varepsilon<1\) and as \(\varepsilon\to 0\), \(\rho_j^\varepsilon\to 0\) and \(v_\varepsilon(r)\to 0\) uniformly on compact subsets of \((0,1]\). Moreover, given any sequence \(\varepsilon_k\to 0\), there is a subsequence \(\varepsilon_{k_i}\) such that \(u_\varepsilon(|x|):= v_\varepsilon(\varepsilon|x|)\) converges to some \(U\) in \(C_{\text{loc}}^1(\mathbb R^N)\) along this subsequence, and \(U=U(|x|)\) is a radial sign changing solution of
\[ -\Delta U+U=f(U) \quad\text{in }\mathbb R^N, \quad U\in H^1(\mathbb R^N) \]
with exactly \(n\) zeros: \(0<\rho_1<\cdots<\rho_n<\infty\), and \(\varepsilon^{-1}\rho_j^\varepsilon\to\rho_j\) along the subsequence \(\varepsilon_{k_i}\). Hence the sharp layers of the sign changing solution \(v_\varepsilon\) are clustered near the origin.
The same result holds if the Neumann boundary condition is replaced by the Dirichlet boundary condition, or if \(B\) is replaced by \(\mathbb R^N\).

MSC:

35J61 Semilinear elliptic equations
35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
58J05 Elliptic equations on manifolds, general theory
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