Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1651–1673 | Cite as

On the Fredholm-type theorems and sign properties of solutions for (pq)-Laplace equations with two parameters

  • Vladimir BobkovEmail author
  • Mieko Tanaka


We consider the Dirichlet problem for the nonhomogeneous equation \(-\Delta _p u -\Delta _q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)\) in a bounded domain, where \(p \ne q\), and \(\alpha , \beta \in \mathbb {R}\) are parameters. We explore assumptions on \(\alpha \) and \(\beta \) that guarantee the resolvability of the considered problem. Moreover, we introduce several curves on the \((\alpha ,\beta )\)-plane allocating sets of parameters for which the problem has or does not have positive or sign-changing solutions, provided f is of a constant sign.


(p, q)-Laplacian Fredholm alternative Existence of solutions Positive solutions Maximum principle Linking method 

Mathematics Subject Classification

35J62 35J20 35P30 35B50 



V. Bobkov was supported by the Grant 18-03253S of the Grant Agency of the Czech Republic and by the project LO1506 of the Czech Ministry of Education, Youth and Sports. M. Tanaka was supported by JSPS KAKENHI Grant Number 15K17577. The authors would like to thank the anonymous referee for valuable remarks and suggestions which helped to improve the manuscript.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and NTIS, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  2. 2.Department of MathematicsTokyo University of ScienceShinjyuku-kuJapan

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