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On a class of semilinear elliptic equations with boundary conditions and potentials which change sign. (English) Zbl 1128.35046
The authors deal with the existence of nontrivial solutions of the following problem: \[ \Delta u=u\text{ in }\Omega\quad \frac{\partial u} {\partial\nu}= \lambda u-W(x)g(u)\text{ on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain set of \(\mathbb{R}^N\), \(N\geq 3\) with smooth boundary \(\partial\Omega\), \(\lambda>0\), \(\lambda\in[0, \lambda_1)\) and \(\frac{\partial}{\partial\gamma}\) is the other normal derivative. Here \(\lambda_1\) is the first eigenvalue of the Steklov problem. As for the \(W\in C(\overline\Omega)\) the authors assume that, \(W\) is different from zero almost everywhere and changes sign, while \(g(n)\) is a continuous and superlinear function. The authors present existence result for (1). The proofs are based on the variational and min-max methods.
35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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