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Multiple solutions for superlinear double phase Neumann problems. (English) Zbl 1483.35101

On a domain \(\Omega \subset \mathbb R^N\) with Lipschitz boundary the authors study the double phase problem \[ \begin{cases} - \text{div}( a(z) |Du(z)|^{r-2} Du(z)) - \Delta_q u(z) + \xi(z) |u(z)|^{q-2} u(z) = f(z, u(z)) &\text{ in } \Omega, \\ \frac{\partial u}{\partial n} = 0 &\text{ on } \partial \Omega, \end{cases} \] with Neumann boundary conditions, for exponents \(1 < q < p <N\). This problem contains several difficulties with respect to a more standard setting, namely that the differential operator is non-homogeneous and that the reaction \(f(z,u)\) is not assumed to satisfy the Ambrosetti-Rabinowitz condition. Under general hypotheses on \(a\), \(\xi\), \(f\), \(p\) and \(q\), using the Nehari manifold method, the authors prove the existence of three nontrivial bounded ground state solutions to the above problem. These solutions are shown to be positive, negative and nodal, respectively.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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