Constant sign solutions for double phase problems with superlinear nonlinearity

Abstract

We study parametric double phase problems involving superlinear nonlinearities without supposing any growth condition. Based on truncation and comparison methods the existence of two constant sign solutions is shown provided the parameter is larger than the first eigenvalue of the p-Laplacian. As a result of independent interest we prove a priori estimates for solutions for a general class of double phase problems with convection term.2010 Mathematics Subject Classification. 35J15, 35J62, 35J92, 35P30.

Least Energy Solutions with Sign Information for Parametric Double Phase Problems

“…Using the Nehari manifold method along the lines of Szulkin-Weth [20] and Lin-Tang [8] (semilinear problems driven by the Dirichlet Laplacian), we show that for all λ < λ 1 (q) problem (P λ ) has at least three nontrivial solutions, all with sign information (positive, negative and nodal (sign-changing)) and with least energy (ground state solutions). Other existence and multiplicity results for different types of double phase equations, can be found in the papers of Colasuonno-Squassina [2], Gasiński-Papageorgiou [3], Gasiński-Winkert [4][5][6], Liu-Dai [9], Papageorgiou-Rȃdulescu-Repovš [11,12], Papageorgiou-Repovš-Vetro [14], Papageorgiou-Vetro-Vetro [15,16], Rȃdulescu [18], Zeng-Bai-Gasiński-Winkert [22,23].…”

A Multiplicity Theorem for Superlinear Double Phase Problems

“…The lack of global regularity theory for double phase problems, leads to a different approach based on the Nehari manifold, as this was developed by Brown-Wu [12], Szulkin-Weth [13] and Willem [14]. Other existence and multiplicity results for double phase problems can be found in the works of Gasiński-Winkert [15] (coercive equations), Colasuonno-Squassina [16], Ge-Wang-Lu [17] (eigenvalue problems), Gasiński-Winkert [18], Papageorgiou-Vetro-Vetro [19] (Robin problems). For the nonlinear problems related to the nonlinear frequency shift phenomena we refer to Kalyabin et al [20] and Sadovnikov et al [21].…”

Convergence analysis for double phase obstacle problems with multivalued convection term

“…Existence and uniqueness results have been recently obtained by several authors. In the case of single-valued equations with or without convection term, we refer to Colasuonno-Squassina [12], Gasiński-Papageorgiou [16,17], Gasiński-Winkert [19][20][21], Liu-Dai [27], Perera-Squassina [39], Papageorgiou-Vetro-Vetro [34,35] and the references therein.…”