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A unified treatment of degenerate nonlocal elliptic problems

L. Gasinski, H. Ramos Quoirin, J. Santos Junior, K. Silva

Abstract

We develop a unified framework for a broad class of nonlocal elliptic problems, encompassing a wide spectrum of nonlocal terms, including the classical Kirchhoff and Carrier-type equations as particular cases, and nonlinearities having sublinear or asymptotically linear growth. By combining the study of a suitable auxiliary problem and fixed-point techniques with careful parameter analysis, we establish existence, non-existence, and multiplicity results for positive solutions. Our method reveals sharp parameter thresholds and provides a comprehensive description of the solution set. Finally, for powerlike nonlinearities (including superlinear and singular ones) we provide a more direct approach, based on homogeneity.

A unified treatment of degenerate nonlocal elliptic problems

Abstract

We develop a unified framework for a broad class of nonlocal elliptic problems, encompassing a wide spectrum of nonlocal terms, including the classical Kirchhoff and Carrier-type equations as particular cases, and nonlinearities having sublinear or asymptotically linear growth. By combining the study of a suitable auxiliary problem and fixed-point techniques with careful parameter analysis, we establish existence, non-existence, and multiplicity results for positive solutions. Our method reveals sharp parameter thresholds and provides a comprehensive description of the solution set. Finally, for powerlike nonlinearities (including superlinear and singular ones) we provide a more direct approach, based on homogeneity.
Paper Structure (10 sections, 22 theorems, 132 equations, 6 figures)

This paper contains 10 sections, 22 theorems, 132 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General Nonlocal Equation and Background
  3. The Powerlike Case
  4. The Auxiliary Problem
  5. Existence of Fixed Points of $P$
  6. The General Case
  7. The Case "$Q$ is Increasing"
  8. Some Bounds for $\lambda_0$
  9. Proofs of the Main Results
  10. The Regular Case

Key Result

Theorem 1.1

Suppose a1, f1 and either g1 or g2 and f2. Then for any $i=0,1,\ldots,k-1$ there exists $0<\lambda_{0,i}<\widetilde{\lambda}_{0,i}<(A_i\lambda_1)/\theta$ such that the following assertions hold: In particular: If, in addition, $Q$ is increasing, then $\lambda_{0,i}=\widetilde{\lambda}_{0,i}$ and for $\lambda=\lambda_{0,i}$ there exists at least one solution $u_i$ of Pg in $(t_i,t_{i+1})$.

Figures (6)

  • Figure 1: $(\alpha_1,\alpha_2)$ is of type $I_{\infty,\infty}$. Here $\lambda=1$ to simplify the figure.
  • Figure 2: $(\alpha_1,\alpha_2)$ is of type $I_{\infty,0}$, while $(\alpha_3,\alpha_4)$ is of type $I_{0,\infty}$.
  • Figure 3: $(\alpha_1,\alpha_2)$ is of type $I_{\infty,0}$, $(\alpha_3,\alpha_4)$ and $(\alpha_5,\alpha_6)$ are of type $I_{0,0}$, while $(\alpha_7,\alpha_8)$ is of type $I_{0,\infty}$.
  • Figure 4: $(\alpha_1,\alpha_2)$ has type $I_{\infty,0}$, while $(\alpha_3,\alpha_4)$ has type $I_{0,\infty}$. Fixed points at $P_1$ and $P_2$.
  • Figure 5: $(\alpha_1,\alpha_2)$ has type $I_{\infty,\infty}$. Fixed points at $P_1$ and $P_2$.
  • ...and 1 more figures

Theorems & Definitions (52)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Theorem 1.4
  • Theorem 1.5
  • ...and 42 more