Competing power problem involving the half Laplacian

Abstract We are concerned with a nonnegative solution to the scalar field equation $$\Delta u + f(u) = 0{\rm in }{\open R}^N,\quad \mathop {\lim }\limits_{|x|\to \infty } u(x) = 0.$$ A classical existence result by Berestycki and Lions [3] considers only the case when f is continuous. In this paper, we are interested in the existence of a solution when f is singular. For a singular nonlinearity f, Gazzola, Serrin and Tang [8] proved an existence result when $f \in L^1_{loc}(\mathbb {R}) \cap \mathrm {Lip}_{loc}(0,\infty )$ with $\int _0^u f(s)\,{\rm d}s < 0$ for small $u>0.$ Since they use a shooting argument for their proof, they require the property that $f \in \mathrm {Lip}_{loc}(0,\infty ).$ In this paper, using a purely variational method, we extend the previous existence results for $f \in L^1_{loc}(\mathbb {R}) \cap C(0,\infty )$ . We show that a solution obtained through minimization has the least energy among all radially symmetric weak solutions. Moreover, we describe a general condition under which a least energy solution has compact support.

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On semiclassical states of a nonlinear Dirac equation

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511001752 ◽

2013 ◽

Vol 143 (4) ◽

pp. 765-790 ◽

Cited By ~ 8

Author(s):

Y. H. Ding ◽

C. Lee ◽

B. Ruf

Keyword(s):

Dirac Equation ◽

Semiclassical Limit ◽

Limit Problem ◽

Energy Solution ◽

Nonlinear Dirac Equation ◽

Least Energy Solution ◽

Semiclassical States ◽

Image Position ◽

Least Energy Solutions ◽

Least Energy

We study the semiclassical limit of the least energy solutions to the nonlinear Dirac equation for x ∈ ℝ3. We prove that the equation has least energy solutions for all ħ > 0 small, and, in addition, that the solutions converge in a certain sense to the least energy solution of the associated limit problem as ħ → 0.

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On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent

Communications on Pure & Applied Analysis ◽

10.3934/cpaa.2013.12.1237 ◽

2013 ◽

Vol 12 (3) ◽

pp. 1237-1241

Author(s):

Futoshi Takahashi ◽

Keyword(s):

Energy Solution ◽

Two Dimensional ◽

Least Energy Solution ◽

Large Exponent ◽

Hénon Equation ◽

Least Energy

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Non-even positive solutions of the Emden–Fowler equations with sign-changing weights

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511001594 ◽

2013 ◽

Vol 143 (3) ◽

pp. 631-642 ◽

Cited By ~ 1

Author(s):

Ryuji Kajikiya

Keyword(s):

Positive Solution ◽

Positive Solutions ◽

Energy Solution ◽

Coefficient Function ◽

Least Energy Solution ◽

Least Energy ◽

Fowler Equation

We study the Emden–Fowler equation whose coefficient function is even in the interval (—1, 1), negative near t = 0 and positive near t = ±1. Then we prove that a least energy solution is not even. Therefore, the equation has an even positive solution and a non-even positive solution.

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Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0011 ◽

2019 ◽

Vol 9 (1) ◽

pp. 496-515 ◽

Cited By ~ 24

Author(s):

Sitong Chen ◽

Xianhua Tang

Keyword(s):

Ground State ◽

Radial Symmetry ◽

Ground State Solution ◽

Bounded Sequence ◽

Energy Solution ◽

Related Literature ◽

Ground State Solutions ◽

Nonlinear Schrödinger ◽

Least Energy Solution ◽

Least Energy

Abstract This paper is dedicated to studying the nonlinear Schrödinger equations of the form $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(u), & x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$ where V ∈ 𝓒1(ℝN, [0, ∞)) satisfies some weak assumptions, and f ∈ 𝓒(ℝ, ℝ) satisfies the general Berestycki-Lions assumptions. By introducing some new tricks, we prove that the above problem admits a ground state solution of Pohožaev type and a least energy solution. These results generalize and improve some ones in [L. Jeanjean, K. Tanka, Indiana Univ. Math. J. 54 (2005), 443-464], [L. Jeanjean, K. Tanka, Proc. Amer. Math. Soc. 131 (2003) 2399-2408], [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our assumptions are “almost” necessary when V(x) ≡ V∞ > 0, moreover, our approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available, or where the ground state solutions of the problem at infinity are not sign definite.

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