# zbMATH — the first resource for mathematics

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.
##### MSC:
 35J60 Nonlinear elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: