Existence of a nontrivial solution for the \((p, q)\)-Laplacian in \(\mathbb{R}^N\) without the Ambrosetti-Rabinowitz condition. (English) Zbl 1308.35114
Summary: In this paper we prove the existence of at least one nonnegative nontrivial weak solution in \(\mathcal{D}^{1, p}(\mathbb{R}^N) \cap \mathcal{D}^{1, q}(\mathbb{R}^N)\) for the equation
\[
- \varDelta_p u - \varDelta_q u + a(x)| u|^{p - 2} u + b(x)| u|^{q - 2} u = f(x, u), x \in \mathbb{R}^N,
\]
where \(1 < q < p < q^\star : = \frac{N q}{N - q}\), \(p < N\), \(\varDelta_m u : = \operatorname{div}(| \nabla u|^{m - 2} \nabla u)\) is the \(m\)-Laplacian operator, the coefficients \(a\) and \(b\) are continuous, coercive and positive functions, and the nonlinearity \(f\) is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti-Rabinowitz condition.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 47J30 | Variational methods involving nonlinear operators |
| 35D30 | Weak solutions to PDEs |
Keywords:
Ambrosetti-Rabinowitz condition; Cerami condition; nontrivial weak solution; \((p,q)\)-LaplacianReferences:
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