Abstract
We study the nonlocal nonlinear problem (-Δ)su=λf(u)inΩ,u=0onRN\\Ω,(Pλ)\\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}$$\\begin{aligned} \\left\\{ \\begin{array}[c]{lll} (-\\Delta )^s u = \\lambda f(u) &{} \\text{ in } \\Omega , \\\\ u=0&{}\\text{ on } \\mathbb {R}^N{\\setminus }\\Omega , \\quad (P_{\\lambda }) \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega is a bounded smooth domain in mathbb {R}^N, N>2s, 0<s<1; f:mathbb {R}rightarrow [0,infty ) is a nonlinear continuous function such that f(0)=f(1)=0 and f(t)sim |t|^{p-1}t as trightarrow 0^+, with 2<p+1<2^*_s; and lambda is a positive parameter. We prove the existence of two nontrivial solutions u_{lambda } and v_{lambda } to (P_{lambda }) such that 0le u_{lambda }< v_{lambda }le 1 for all sufficiently large lambda . The first solution u_{lambda } is obtained by applying the Mountain Pass Theorem, whereas the second, v_{lambda }, via the sub- and super-solution method. We point out that our results hold regardless of the behavior of the nonlinearity f at infinity. In addition, we obtain that these solutions belong to L^{infty }(Omega ).
Highlights
This paper concerns with the existence of nonnegative solutions of the following nonlocal nonlinear elliptic problem (− )su = λ f (u) in, u=0 on RN \, (1.1)where is a bounded smooth domain in RN, N > 2s, 0 < s < 1; f : R → [0, ∞)is a nonlinear continuous function such that f (0) = f (1) = 0 and f (t) ∼ |t|p−1t as t → 0+, with 1 < p N +2s N −2s = 2s∗
We prove the existence of two nontrivial solutions uλ and vλ to (Pλ) such that 0 ≤ uλ < vλ ≤ 1 for all sufficiently large λ
The document is organized as follows: in Sect. 2 we offer a brief review of the fractional spaces of Sobolev in the context of our problem and recall some useful results
Summary
This paper concerns with the existence of nonnegative solutions of the following nonlocal nonlinear elliptic problem (− )su = λ f (u) in , u=0 on RN \ ,. It is known that its behavior at zero and/or infinity can get to play a crucial role in the existence question of solutions. This study has been extended to problems where the Laplacian was replaced by the p-Laplacian [13,14] or Pucci’s operators [1,20] In all these works, it was shown that there exist two positive solutions for sufficiently large λ and by assuming some additional conditions on f. To put into perspective our result, throughout this paper we consider f : R → [0, ∞) being a continuous function that verifies the following conditions:.
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