Abstract
In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem: (−Δ)12u+u+(ln|⋅|∗|u|2)=f(u)+μ|u|−γ−1u,in R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\begin{aligned} (-\\Delta )^{\\frac{1}{2}}u+u+\\bigl(\\ln \\vert \\cdot \\vert * \\vert u \\vert ^{2}\\bigr)&=f(u)+\\mu \\vert u \\vert ^{- \\gamma -1}u,\\quad \\text{in }\\mathbb{R}, \\end{aligned} \\end{aligned}$$ \\end{document} where mu >0, (*) is the convolution operation between two functions, 0<gamma <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.
Highlights
The main objective of this paper is to establish the existence and multiplicity of nontrivial solutions for the following problem: )2 u + u + ln | · | ∗ |u|2= f (u) + μ|u|–γ –1u in R, (1.1)where μ > 0, (∗) is the convolution operation between two functions, 0 < γ < 1
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
Using subcritical or critical polynomial growth of the function f is quite common in the literature pertaining to problems on elliptic PDEs
Summary
The main objective of this paper is to establish the existence and multiplicity of nontrivial solutions for the following problem:. Since the setting is periodic, the global Palais–Smale condition can fail due to the invariance of the functional under the Z2-translations To tackle this problem, Du and Weth [20] proved the existence of a mountain pass solution and a ground state solution for local problem (1.1) but without the singular term in the case when V (x) ≡ α > 0, 2 < p < 4, and f (u) = |u|p–2u. Alves and Figueiredo [3] proved the existence of a ground state solution to problem (1.1) without the singular term but with a nonlinearity of the Moser–Trudinger type. We give an Appendix to the proofs of all results which have been used in the proof of the main theorem
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