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Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent

Souvik Bhowmick, Sekhar Ghosh, Vishvesh Kumar

Abstract

This paper addresses a class of elliptic problems involving the superposition of nonlinear fractional operators with the critical Sobolev exponent in the sublinear regimes. We establish the existence of infinitely many nontrivial weak solutions using a variational framework combining a truncation argument with the notion of genus. A central part of our analysis is the verification of the Palais--Smale (PS) condition for the associated energy functional for every $q \in (1, p_{s_\sharp}^*)$, despite the challenges posed by the lack of compactness due to the critical exponent. The results obtained in the paper are new even in the classical case $p = 2$, highlighting the broader applicability of the methods developed here.

Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent

Abstract

This paper addresses a class of elliptic problems involving the superposition of nonlinear fractional operators with the critical Sobolev exponent in the sublinear regimes. We establish the existence of infinitely many nontrivial weak solutions using a variational framework combining a truncation argument with the notion of genus. A central part of our analysis is the verification of the Palais--Smale (PS) condition for the associated energy functional for every , despite the challenges posed by the lack of compactness due to the critical exponent. The results obtained in the paper are new even in the classical case , highlighting the broader applicability of the methods developed here.

Paper Structure

This paper contains 4 sections, 20 theorems, 113 equations.

Table of Contents

  1. Introduction and main results
  2. Functional Analytic setting for the superposition operator
  3. Existence of infinitely many solutions
  4. Important examples and applications

Key Result

Theorem 1.1

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}.$ Let $\mu=\mu^{+}-\mu^{-}$ with $\mu^{+}$ and $\mu^{-}$ satisfy conditions measure 1-measure 3. Assume that $s_{\sharp}$ is given as in measure 4. Let $1 < p < \frac{N}{s_\sharp}$ and $1 < q < p$. Then, there exist $\lambda_*>0$ and $\kappa_*

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • ...and 29 more