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
\[ \begin{cases} -\Delta u+u= \lambda f(x)|u|^{q-1}+ h(x)|u|^{p-1} &\text{ in }\Omega,\\ u>0 &\text{ in }\Omega,\\ u=0 &\text{ on }\partial\Omega,\end{cases} \]
are established, where \(\Omega=\Omega'\times\mathbb R\), \(\Omega'\subset\mathbb R^{N-1}\) is bounded smooth domain and \(N\geq2\). Here \(1<q<2<p<2^*\) \((2^*= \frac{2N}{N-2}\) if \(N\geq 3\), \(2^*=\infty\) 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.
\[ \begin{cases} -\Delta u+u= \lambda f(x)|u|^{q-1}+ h(x)|u|^{p-1} &\text{ in }\Omega,\\ u>0 &\text{ in }\Omega,\\ u=0 &\text{ on }\partial\Omega,\end{cases} \]
are established, where \(\Omega=\Omega'\times\mathbb R\), \(\Omega'\subset\mathbb R^{N-1}\) is bounded smooth domain and \(N\geq2\). Here \(1<q<2<p<2^*\) \((2^*= \frac{2N}{N-2}\) if \(N\geq 3\), \(2^*=\infty\) 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 | Variational methods for second-order elliptic equations |
| 35B20 | Perturbations in context of PDEs |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
Keywords:
multiple positive solutions; concave-convex nonlinearities; Nehari manifold; sign-changing weight functionsReferences:
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