Multiple solutions of double phase variational problems with variable exponent

Abstract

Abstract This paper deals with the existence of multiple solutions for the quasilinear equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mrow><m:mo>-</m:mo><m:mrow><m:mrow><m:mi>div</m:mi><m:mo>⁡</m:mo><m:mi>𝐀</m:mi></m:mrow><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mrow><m:mo>∇</m:mo><m:mo>⁡</m:mo><m:mi>u</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:mrow><m:mo>+</m:mo><m:mrow><m:msup><m:mrow><m:mo>|</m:mo><m:mi>u</m:mi><m:mo>|</m:mo></m:mrow><m:mrow><m:mrow><m:mi>α</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>-</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi>u</m:mi></m:mrow></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>f</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>u</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo> </m:mo><m:mrow><m:mtext>in </m:mtext><m:msup><m:mi>ℝ</m:mi><m:mi>N</m:mi></m:msup><m:mtext>,</m:mtext></m:mrow></m:mrow></m:mrow></m:math> {-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}} which involves a general variable exponent elliptic operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>𝐀</m:mi></m:math> {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mrow><m:mo>|</m:mo><m:mi>ξ</m:mi><m:mo>|</m:mo></m:mrow><m:mrow><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>-</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi>ξ</m:mi></m:mrow></m:math> {|\xi|^{q(x)-2}\xi} for small <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>|</m:mo><m:mi>ξ</m:mi><m:mo>|</m:mo></m:mrow></m:math> {|\xi|} and like <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msup><m:mrow><m:mo>|</m:mo><m:mi>ξ</m:mi><m:mo>|</m:mo></m:mrow><m:mrow><m:mrow><m:mi>p</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>-</m:mo><m:mn>2</m:mn></m:mrow></m:msup><m:mo>⁢</m:mo><m:mi>ξ</m:mi></m:mrow></m:math> {|\xi|^{p(x)-2}\xi} for large <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>|</m:mo><m:mi>ξ</m:mi><m:mo>|</m:mo></m:mrow></m:math> {|\xi|} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>1</m:mn><m:mo>&lt;</m:mo><m:mrow><m:mi>α</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mo>⋅</m:mo><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>≤</m:mo><m:mrow><m:mi>p</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mo>⋅</m:mo><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>&lt;</m:mo><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mo>⋅</m:mo><m:mo>)</m:mo></m:mrow></m:mrow><m:mo>&lt;</m:mo><m:mi>N</m:mi></m:mrow></m:math> {1&lt;\alpha(\,\cdot\,)\leq p(\,\cdot\,)&lt;q(\,\cdot\,)&lt;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 [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>ℝ</m:mi><m:mi>N</m:mi></m:msup></m:math> \mathbb{R}^{N} via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents p and q are constant, to the case where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>p</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mo>⋅</m:mo><m:mo>)</m:mo></m:mrow></m:mrow></m:math> {p(\,\cdot\,)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>q</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mo>⋅</m:mo><m:mo>)</m:mo></m:mrow></m:mrow></m:math> {q(\,\cdot\,)} are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.