Multiple solutions of double phase variational problems with variable exponent

Abstract

This paper deals with the existence of multiple solutions for the quasilinear equation $-\mathrm{div}\mathbf{A}(x,\nabla u)|u{|}^{\alpha (x)-2}u=f(x,u)$ in ${\mathbb{R}}^N$, which involves a general variable exponent elliptic operator $\mathbf{A}$ in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like $|\xi {|}^{q(x)-2}\xi$ for small $|\xi |$ and like $|\xi {|}^{p(x)-2}\xi$ for large $|\xi |$, where $1<\alpha (\cdot )\le p(\cdot )<q(\cdot )<N$. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Our results extend the previous works Azzollini, d'Avenia, and Pomponio (2014) and Chorfi and Rădulescu (2016), from the case when exponents $p$ and $q$ are constant, to the case when $p(\cdot )$ and are functions. We also substantially weaken some of their hypotheses overcome the lack of compactness by using the weighting method.

References in corpus (2)

• Stationary waves of Schrödinger-type equations with variable exponent
• On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting

Cited by in corpus (5)

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• Double phase problems with variable growth
• The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth
• Existence of Weak Solutions for $p(.)$-Laplacian Equation via Compact Embeddings of the Double Weighted Variable Exponent Sobolev Spaces
• Fractional double phase Robin problem involving variable order-exponents without Ambrosetti-Rabinowitz condition