Journal
JOURNAL OF GEOMETRIC ANALYSIS
Volume 32, Issue 9, Pages -Publisher
SPRINGER
DOI: 10.1007/s12220-022-00983-3
Keywords
Fractional double-phase problem; Positive ground states; Concentration; Multiplicity
Categories
Funding
- Natural Science Foundation of Hunan Province [2021JJ30189, 2022JJ30200]
- Key project of Scientific Research Project of Department of Education of Hunan Province [21A0387]
- China Scholarship Council [201908430218, 201908430219]
- Funding scheme for Young Backbone Teachers of universities in Hunan Province (Hunan Education Notification (2018)) [574]
- Funding scheme for Young Backbone Teachers of universities in Hunan Province (Hunan Education Notification (2020)) [43]
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This paper deals with a singularly perturbed fractional double-phase problem with unbalanced growth and competing potentials. By utilizing topological and variational methods, the existence and concentration phenomena of positive solutions are established, as well as the multiplicity result dependent on the topology of the set where the potentials attain their extrema.
This paper is concerned with the following singularly perturbed fractional double-phase problem with unbalanced growth and competing potentials {epsilon(ps) (-Delta)(p)(s)u + epsilon(qs) (-Delta)(q)(s) u + V(x) (vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(q-2)u) = W(x)g(u), in R-N, u is an element of W-s,W- p (R-N) boolean AND W-s,W- q(R-N), u > 0, where s is an element of (0, 1), 2 <= p < q < N/s, (-Delta)(t)(s) with t is an element of (p,q), is the fractional t-Laplacian operator, epsilon > 0 is a small parameter, V is the absorption potential, W is the reaction potential and g is the reaction term with subcritical growth. Assume that the potentials V, W, and the nonlinearity g satisfy some natural conditions, applying topological and variational methods, we establish the existence and concentration phenomena of positive solutions for epsilon > 0 sufficiently small as well as the multiplicity result depended on the topology of the set where V attains its global minimum and W attains its global maximum. Finally, we also obtain the nonexistence result of ground state solutions under suitable conditions.
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