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Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains. (English) Zbl 1173.35498
Summary: Existence and multiplicity results to the following Dirichlet problem $$\cases -\Delta u+u= \lambda f(x)|u|^{q-1}+ h(x)|u|^{p-1} &\text{ in }\Omega,\\ u>0 &\text{ in }\Omega,\\ u=0 &\text{ on }\partial\Omega,\endcases$$ are established, where $\Omega=\Omega'\times\Bbb R$, $\Omega'\subset\Bbb R^{N-1}$ is bounded smooth domain and $N\ge2$. Here $1<q<2<p<2^*$ $(2^*= \frac{2N}{N-2}$ if $N\ge 3$, $2^*=\infty$ if $N=2)$ $\lambda$ is a positive real parameter, the function $f$, among other conditions, can possibly change sign in $\Omega$, and the function $h$ satisfies suitable conditions. The study is based on the comparison of energy levels on Nehari manifold.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Second order elliptic equations, variational methods 35B20 Perturbations (PDE) 35D05 Existence of generalized solutions of PDE (MSC2000)
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##### References:
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