Positive solutions for nonlinear Schrödinger–Kirchhoff equations in R3

Nontrivial Solutions for 4-Superlinear Schrödinger–Kirchhoff Equations with Indefinite Potentials

“…Problem (1) has been studied extensively by many researchers. Some interesting studies by variational methods can be found in, for example, [3][4][5][6][7][8][9][10][11][12][13] and references therein. Under suitable conditions, it is well known that weak solutions to (1) correspond to critical points of the energy functional Φ :…”

A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator

“…where 0 < α ≤ 1, 1 < β ≤ 2. In the past twenty years, there has been a growing interest in the profit derived from advancing space theories [18][19][20], regular theories [21][22][23][24][25], operator methods [26][27][28][29][30], iterative techniques [31][32][33], the moving sphere method [34], critical point theories [35][36][37][38], and tempered fractional calculus. This surge in attention has not only propelled the rapid progress of these disciplines, but has also spurred corresponding contributions across various fields.…”

The extreme solutions for a σ‐Hessian equation with a nonlinear operator

“…Various nonlinear analysis theories and methods may be used to study the 𝜎-Hessian equation, such as the spaces theories [12][13][14][15][16][17][18][19][20][21], smoothness theories [22][23][24][25][26][27], operator theories [28][29][30][31], fixed point theorems [32][33][34][35][36], sub-super solution methods [37][38][39], monotone iterative techniques [40,41], and the variational method [42][43][44]. For example, by adopting the sub-super solution method, Zhang et al [37] recently established the interval of the eigenvalue in which the existence of solutions for the following singular augmented 𝜎-Hessian equation is guaranteed…”

“…Various nonlinear analysis theories and methods may be used to study the $$$ \sigma $$$‐Hessian equation, such as the spaces theories [12–21], smoothness theories [22–27], operator theories [28–31], fixed point theorems [32–36], sub‐super solution methods [37–39], monotone iterative techniques [40, 41], and the variational method [42–44]. For example, by adopting the sub‐super solution method, Zhang et al [37] recently established the interval of the eigenvalue in which the existence of solutions for the following singular augmented $$$ \sigma $$$‐Hessian equation is guaranteed where $$$ {B}_1&amp;amp;#x0003D;\left\{x\in {\mathrm{\mathbb{R}}}&amp;amp;#x0005E;M:&amp;amp;#x0007C;x&#x0007C;&amp;lt;1\right\} $$$, …”