A multiplicity theorem for parametric superlinear (p,q)-equations

Abstract

We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.

Ground state solution for a periodic p\&q-Laplacain equation involving critical growth without the Ambrosetti-Rabinowitz condition

“…In recent years, there are extensive bibliographies in the study of the quasilinear equation of the p&q-Laplacian type, see e.g. [5,10,11,13,13,20,1,7,25,26,8,16,24,4,12,22,19,21,9,3,17,19,2] and the references therein.…”

Ground state solution for a periodic p&q‐Laplacian equation involving critical growth without the Ambrosetti–Rabinowitz condition

“…To search for the ground state solutions, let's introduce the ground state energy and Nehari manifold corresponding to $$$ {J}_K^{\lambda } $$$, $$$$ {m}_K^{\lambda}\triangleq \underset{u\in {\mathcal{N}}_K^{\lambda }}{\operatorname{inf}}{J}_K^{\lambda }(u), $$$$ where $$$$ {\mathcal{N}}_K^{\lambda }=\left\{u\in E\backslash \left\{0\right\}:\Big\langle {\left({J}_K^{\lambda}\right)}^{\prime }(u),u\Big\rangle =0\right\}. $$$$ In recent years, there are extensive bibliographies in the study of the quasilinear equation of the p&q‐Laplacian type, see, for example, previous works [1–4, 6–24] and the references therein.…”

“…In recent years, there are extensive bibliographies in the study of the quasilinear equation of the p&q-Laplacian type, see, for example, previous works [1][2][3][4][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein. Cherfils-Il'yasov [2] obtained the existence and nonexistence results for the problem…”