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Existence and nonexistence of infinitely many solutions to elliptic problems with oscillating nonlinearities

Antonio J. Martínez Aparicio, Clara Torres-Latorre

Abstract

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-Δ_p u = λf(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function with a sequence of positive zeros converging to zero or diverging to infinity. Under a growth condition on the primitive $F(s) = \int_0^s f(t)dt$, we establish ranges of the parameter $λ$ for which infinitely many small or large solutions exist, as well as ranges where no bifurcation from zero or infinity can occur. The existence result is obtained via variational methods for a general class of divergence form operators, while the nonexistence result is established both for the $p$-Laplacian and for uniformly elliptic operators in non-divergence form via an ODE argument.

Existence and nonexistence of infinitely many solutions to elliptic problems with oscillating nonlinearities

Abstract

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem in a bounded domain with Dirichlet boundary conditions, where is a continuous function with a sequence of positive zeros converging to zero or diverging to infinity. Under a growth condition on the primitive , we establish ranges of the parameter for which infinitely many small or large solutions exist, as well as ranges where no bifurcation from zero or infinity can occur. The existence result is obtained via variational methods for a general class of divergence form operators, while the nonexistence result is established both for the -Laplacian and for uniformly elliptic operators in non-divergence form via an ODE argument.
Paper Structure (11 sections, 7 theorems, 69 equations)

This paper contains 11 sections, 7 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Plan of the paper
  5. Setting
  6. Divergence form operators
  7. Non-divergence form operators
  8. Existence of infinitely many solutions
  9. Infinitely many small solutions
  10. Infinitely many large solutions
  11. Nonexistence of bifurcation points

Key Result

Theorem 1.1

Let $\ell\in \{0,\infty\}$, and let $f\in C(\mathbb{R})$ be a function with $f(0)\geq 0$. Suppose that eq:hyp_zeros holds, and that Then, there exists $\overline{\lambda} \in [0,\infty)$ such that, for any $\lambda>\overline{\lambda}$, problem eq:Pb_pLap has a sequence $\{u_n\}$ of nonnegative (and nontrivial) bounded solutions such that $\|u_n\|_{L^{\infty}(\Omega)} \to \ell$. Furthermore, if $L

Theorems & Definitions (18)

  • Theorem 1.1: Infinitely many solutions
  • Theorem 1.3: Nonexistence of bifurcation points
  • Corollary 1.4
  • Definition 2.1
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 8 more