Ground state sign-changing solutions for semilinear Dirichlet problems

Xiaoyan Lin,Xianhua Tang

doi:10.1186/s13661-018-0973-3

Abstract

In the present paper, we consider the existence of ground state sign-changing solutions for the semilinear Dirichlet problem \t\t\t0.1{−△u+λu=f(x,u),x∈Ω;u=0,x∈∂Ω,\\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}$$ \\left \\{ \\textstyle\\begin{array}{l@{\\quad}l} -\\triangle u+\\lambda u=f(x, u), & \\hbox{$x\\in\\Omega$;} \\\\ u=0, & \\hbox{$x\\in\\partial\\Omega$,} \\end{array}\\displaystyle \\right . $$\\end{document} where Omegasubsetmathbb{R}^{N} is a bounded domain with a smooth boundary ∂Ω, lambda>-lambda_{1} is a constant, lambda_{1} is the first eigenvalue of (-triangle, H_{0}^{1}(Omega)), and fin C(Omegatimesmathbb{R}, mathbb{R}). Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function tmapsto f(x, t)/|t| is non-decreasing on (-infty, 0)cup(0, infty) for every xinOmega, we prove that (0.1) possesses one ground state sign-changing solution, which has precisely two nodal domains. Our results improve and generalize some existing ones.

Highlights

Let ⊂ RN be a bounded domain with a smooth boundary ∂
Under some standard growth assumptions on f and a weak version of Nehari type monotonicity condition that the function t → f (x, t)/|t| is non-decreasing on (–∞, 0) ∪ (0, ∞) for every x ∈, we prove that (0.1) possesses one ground state sign-changing solution, which has precisely two nodal domains
In this paper we are concerned with the existence of sign-changing solutions of the semilinear Dirichlet problem

Summary

Introduction

Let ⊂ RN be a bounded domain with a smooth boundary ∂. Bartsch and Weth [6] proved that (1.1) has a least energy sign-changing solution u , i.e., (u ) = m0, which has precisely two nodal domains under (F1), (F2) and the following assumptions:. Under (F1), (F2), and (Ne), Bartsch and Weth [6] proved that every weak solution u ∈ M of (1.1) with 0 < (u) ≤ m0 has precisely two nodal domains, while Bartsch et al [17] showed that every minimizer of on M is a critical point of , a sign-changing solution of (1.1) with precisely two nodal domains. Similar to the proof of [17, Proposition 3.1], we can prove the following lemma. In view of Lemma 2.4, there exists a pair (sn, tn) of positive numbers such that snv+ε + tnv–ε ∈.

Conclusion

Methods