We investigate a nonlinear nonlocal eigenvalue problem involving the fractional (p, q)-Laplace operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(-\Delta )_p^{s_1}+(-\Delta )_q^{s_2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </msubsup> <mo>+</mo> <msubsup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>q</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s_1,s_2\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>; <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p,q\in (1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and subject to Dirichlet boundary conditions in an open bounded set of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear nonlocal eigenvalue problem, and show the existence of multiple solutions of the nonlinear nonlocal problem using variational methods and some topological techniques.

Bifurcation results and multiple solutions for the fractional (p, q)-Laplace operators

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2026年2月17日

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Emmanuel Wend-Benedo Zongo,
Pierre Aime Feulefack

摘要

We investigate a nonlinear nonlocal eigenvalue problem involving the fractional (p, q)-Laplace operators $(-\Delta )_p^{s_1}+(-\Delta )_q^{s_2}$ ${(- Δ)}_{p}^{s_{1}} + {(- Δ)}_{q}^{s_{2}}$ with $s_1,s_2\in (0,1)$ $s_{1}, s_{2} \in (0, 1)$ ; $p,q\in (1,\infty )$ $p, q \in (1, \infty)$ and subject to Dirichlet boundary conditions in an open bounded set of $\mathbb {R}^N$ $R^{N}$ . We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear nonlocal eigenvalue problem, and show the existence of multiple solutions of the nonlinear nonlocal problem using variational methods and some topological techniques.