Existence and multiplicity results for double phase problem

Existence and uniqueness results for double phase problems with convection term

“…The corresponding eigenvalue problem of the double phase operator with Dirichlet boundary condition was analyzed by Colasuonno-Squassina [8] who proved the existence and properties of related variational eigenvalues. By applying variational methods, Liu-Dai [19] treated double phase problems and proved existence and multiplicity results.…”

Ground state sign-changing solutions for a class of double-phase problem in bounded domains

“…It arises from the nonlinear elasticity theory, strongly anisotropic materials, Lavrentiev's phenomenon, and so on (see [2][3][4][5]). The study on double-phase problems attracts more and more interest in recent years, and many results have been obtained [1,[6][7][8][9][10]. More precisely, the research is related to the energy functional…”

“…x ∈ Ω and all t ∈ R for some s ∈ (p, q) and c > 0. Recently, Liu and Dai [1] investigated the sign-changing ground state solution of (P) under (h 1 ), (h 2 ), (h 3 ), and (h 4 ) the function t → f (x,t) |t| q-1 is strictly increasing on (-∞, 0) ∪ (0, +∞). Additionally, Liu and Dai [9] also obtained the existence of at least three ground state solutions of (P) by using the strong maximum principle for the homogeneous doublephase problem.…”

“…It is a well-known consequence of (h 4 ) that there is unique t u > 0 such that t u u ∈ N 0 for every u ∈ W 1,H 0 (Ω)\{0}, which implies that ϕ has at most one minimizer on M 0 . Moreover, (h 4 ) plays a crucial role in [1]. In fact, condition (h 4 ) implies that every minimizer of ϕ on M 0 is a critical point.…”

Constant sign solutions for double phase problems with superlinear nonlinearity

“…, where ·, · H is the duality pairing between W 1,H 0 (Ω) and its dual space W 1,H 0 (Ω) * . The properties of the operator A : W 1,H 0 (Ω) → W 1,H 0(Ω) * are summarized in the following proposition, see Liu-Dai[13]. The operator A defined by (2.5) is bounded, continuous, monotone (hence maximal monotone) and of type (S + ).…”