This paper is concerned with a system of nonlinear elliptic equations driven by <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(p(\cdot ), q(\cdot ))$</EquationSource> </InlineEquation>-Laplacian type operators with variable exponents, subject to homogeneous Neumann boundary conditions. The operators of the form <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mo>−</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mrow> <mi>I</mi> </msubsup> <mo>−</mo> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow> <msub> <mi>q</mi> <mi>i</mi> </msub> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mrow> <mi>I</mi> </msubsup> </math></EquationSource> <EquationSource Format="TEX">$-\Delta _{p_{i}(\cdot )}^{I} - \Delta _{q_{i}(\cdot )}^{I}$</EquationSource> </InlineEquation> extend the classical <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$p(\cdot )$</EquationSource> </InlineEquation>-Laplacian by involving two distinct variable exponents, <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$p_{i}(\cdot )$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msub> <mi>q</mi> <mi>i</mi> </msub> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$q_{i}(\cdot )$</EquationSource> </InlineEquation>. thereby giving rise to the so-called double-phase phenomenon, This structure gives rise to a nonstandard growth phenomenon, whose behavior varies across the domain Ω. The model further incorporates the combined effects of a coercive potential <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\lambda _{i}(x)$</EquationSource> </InlineEquation>, a spatial weight <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>β</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\beta (x)$</EquationSource> </InlineEquation>, and nonlinearities f and g.By applying variational methods, in particular the mountain pass theorem together with appropriate truncation techniques, we establish the existence of at least one nontrivial weak solution. Moreover, under additional symmetry assumptions on the nonlinearities, we prove the existence of an unbounded sequence of weak solutions. Our analysis also relies on an abstract variational principle due to B. Ricceri.The results obtained here extend and complement several recent contributions in the literature. In particular, these results provide new existence and multiplicity results for systems with <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(p(\cdot ), q(\cdot ))$</EquationSource> </InlineEquation>-Laplacian type operators, thus generalizing the constant exponent framework to a more flexible and physically relevant setting.

Infinitely many solutions for a Neumann elliptic system with $(p(\cdot ), q(\cdot ))$-Laplacian type operator

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2026年4月23日

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Ahmed Ahmed,
Khaled Kefi

摘要

This paper is concerned with a system of nonlinear elliptic equations driven by $(p (\cdot), q (\cdot))$ $(p(\cdot ), q(\cdot ))$ -Laplacian type operators with variable exponents, subject to homogeneous Neumann boundary conditions. The operators of the form $- Δ_{p_{i} (\cdot)}^{I} - Δ_{q_{i} (\cdot)}^{I}$ $-\Delta _{p_{i}(\cdot )}^{I} - \Delta _{q_{i}(\cdot )}^{I}$ extend the classical $p (\cdot)$ $p(\cdot )$ -Laplacian by involving two distinct variable exponents, $p_{i} (\cdot)$ $p_{i}(\cdot )$ and $q_{i} (\cdot)$ $q_{i}(\cdot )$ . thereby giving rise to the so-called double-phase phenomenon, This structure gives rise to a nonstandard growth phenomenon, whose behavior varies across the domain Ω. The model further incorporates the combined effects of a coercive potential $λ_{i} (x)$ $\lambda _{i}(x)$ , a spatial weight $β (x)$ $\beta (x)$ , and nonlinearities f and g.

By applying variational methods, in particular the mountain pass theorem together with appropriate truncation techniques, we establish the existence of at least one nontrivial weak solution. Moreover, under additional symmetry assumptions on the nonlinearities, we prove the existence of an unbounded sequence of weak solutions. Our analysis also relies on an abstract variational principle due to B. Ricceri.

The results obtained here extend and complement several recent contributions in the literature. In particular, these results provide new existence and multiplicity results for systems with $(p (\cdot), q (\cdot))$ $(p(\cdot ), q(\cdot ))$ -Laplacian type operators, thus generalizing the constant exponent framework to a more flexible and physically relevant setting.

Infinitely many solutions for a Neumann elliptic system with \((p(\cdot ), q(\cdot ))\)-Laplacian type operator

摘要