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Constant sign and nodal solutions for superlinear double phase problems. (English) Zbl 1478.35118

In this paper the authors study the following double phase problem \begin{align*} \begin{cases} -\text{div}\left(a(z)|Du|^{p-2}Du+|Du|^{q-2}Du\right)=f(z,u)& \text{in }\Omega,\\ u=0& \text{on }\partial\Omega, \end{cases} \end{align*} where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with Lipschitz boundary \(\partial\Omega\), \(1<q<p\), \(0\leq a(\cdot)\in C^{0,1}(\overline{\Omega})\) and \(f\colon\Omega\times\mathbb{R}\to\mathbb{R}\) is a measurable function such that \(f(z,\cdot)\) is locally Lipschitz and it has \((p-1)\)-superlinear growth near \(\pm\infty\). Applying variational tools and the Nehari manifold method, the authors show that the problem has at least three nontrivial weak solutions, whereby one is positive, one is negative and the third one is a nodal solution. In addition, the nodal solution has exactly two nodal domains.

MSC:

35J60 Nonlinear elliptic equations
Full Text: DOI

References:

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