Existence of Continuous Eigenvalues for a Class of Parametric Problems Involving the (p,2)$(p,2)$-Laplacian Operator

Abstract

We discuss a parametric eigenvalue problem, where the differential operator is of (p, 2)-Laplacian type. We show that, when p 2, the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to p > 2 and p < 2, and the methods that are applied are variational. In the former case, the direct method is applied, whereas in the latter case, the fibering method of Pohozaev is used. We will also discuss a priori bounds and regularity of the eigenfunctions. In particular, we will show that, when the eigenvalue tends towards the end point of the half line, the supremum norm of the corresponding eigenfunction tends to zero in the case of p > 2, and to infinity in the case of p < 2.

Perturbed eigenvalue problems: an overview

“…We would like to point out that similar results with those obtained in Theorem 3.1 were obtained by Bhattacharya, Emamizadeh, & Farjudian in [3] and by the first author of this paper in [8, Theorem 1] but for a class of anisotropic differential operators.…”

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 $$$ {J}_K&amp;amp;amp;#x0005E;{\lambda } $$$, $$$$ {m}_K&amp;amp;amp;#x0005E;{\lambda}\triangleq \underset{u\in {\mathcal{N}}_K&amp;amp;amp;#x0005E;{\lambda }}{\operatorname{inf}}{J}_K&amp;amp;amp;#x0005E;{\lambda }(u), $$$$ where $$$$ {\mathcal{N}}_K&amp;amp;amp;#x0005E;{\lambda }&amp;amp;amp;#x0003D;\left\{u\in E\backslash \left\{0\right\}:\Big\langle {\left({J}_K&amp;amp;amp;#x0005E;{\lambda}\right)}&amp;amp;amp;#x0005E;{\prime }(u),u\Big\rangle &amp;amp;amp;#x0003D;0\right\}. $$$$ 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…”