scholarly journals Ground State Solution for a Kirchhoff Problem with Exponential Critical Growth

2015 ◽  
Vol 84 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Giovany M. Figueiredo ◽  
Uberlandio B. Severo
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
Critical Growth ◽  
State Solution ◽  
Exponential Critical Growth ◽  
Kirchhoff Problem

scholarly journals Elliptic problem driven by different types of nonlinearities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Debajyoti Choudhuri ◽  
Dušan D. Repovš
Keyword(s):  
Nontrivial Solution ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Different Types ◽  
Exponential Critical Growth

AbstractIn this paper we establish the existence and multiplicity of nontrivial solutions to the following problem: $$\begin{aligned} \begin{aligned} (-\Delta )^{\frac{1}{2}}u+u+\bigl(\ln \vert \cdot \vert * \vert u \vert ^{2}\bigr)&=f(u)+\mu \vert u \vert ^{- \gamma -1}u,\quad \text{in }\mathbb{R}, \end{aligned} \end{aligned}$$ ( − Δ ) 1 2 u + u + ( ln | ⋅ | ∗ | u | 2 ) = f ( u ) + μ | u | − γ − 1 u , in  R , where $\mu >0$ μ > 0 , $(*)$ ( ∗ ) is the convolution operation between two functions, $0<\gamma <1$ 0 < γ < 1 , f is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity f is of exponential critical growth.

scholarly journals Lions-type theorem of the p-Laplacian and applications

2021 ◽  
Vol 10 (1) ◽  
pp. 1178-1200
Author(s):  
Yu Su ◽  
Zhaosheng Feng
Keyword(s):  
Elliptic Equation ◽  
Ground State ◽  
Quasilinear Elliptic Equation ◽  
Type Theorem ◽  
Ground State Solution ◽  
Critical Growth ◽  
State Solution ◽  
Quasilinear Elliptic

Abstract In this article, our aim is to establish a generalized version of Lions-type theorem for the p-Laplacian. As an application of this theorem, we consider the existence of ground state solution for the quasilinear elliptic equation with the critical growth.

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