Mixed double phase equations with local and nonlocal operators

Abstract: In this paper, we study a new class of mixed double phase problems that combine local and nonlocal operators. We consider two different models. The first model is driven by the fractional $p$-Laplacian together with a local double phase operator, while the second model involves the local $p$-Laplacian coupled with a fractional double phase operator. In order to describe the interaction between local and nonlocal effects within the double phase framework, we introduce an appropriate variational setting based on classical and fractional Musielak-Orlicz Sobolev spaces. Within this setting, we establish several existence and multiplicity results for weak solutions by means of variational and topological techniques. In particular, for the problem driven by the fractional $p$-Laplacian and a local double phase operator, we prove the existence of a nonnegative solution using the Nehari manifold method in the presence of concave-convex nonlinearities. We also investigate the associated Brezis-Nirenberg type problem and obtain the existence of infinitely many solutions via genus theory. For the problem governed by the local $p$-Laplacian and a fractional double phase operator, we show the existence of at least two nontrivial constant sign solutions by exploiting the variational structure of the associated energy functional. Furthermore, in the subcritical case, we prove the existence of a least energy sign-changing solution by combining the Poincaré-Miranda existence theorem with the quantitative deformation lemma.

• The paper develops Musielak–Orlicz energy spaces for two complementary mixed local–nonlocal double phase models and establishes reflexivity, norm equivalence, and compact embeddings needed for variational analysis.
• The paper proves small-parameter existence results, including a negative-energy nonnegative solution for a concave–convex problem and infinitely many negative-energy solutions for a critical Brezis–Nirenberg variant via genus theory.
• The paper obtains three solutions for a converse mixed operator—a positive, a negative, and a least-energy sign-changing solution—without imposing the Ambrosetti–Rabinowitz condition, using Cerami compactness and nodal constraints.

Problem and motivation

The paper studies two classes of quasilinear boundary value problems that couple local and nonlocal diffusion within a double phase framework. The first model combines the fractional $p$-Laplacian with the classical (local) double phase operator $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$, driven by concave–convex nonlinearities with positive weights; its critical variant is a Brezis–Nirenberg type problem with exponent $p^* = Np/(N-p)$. The second model reverses the roles: the local $p$-Laplacian is coupled with a fractional double phase operator whose Gagliardo-type kernel carries a weight $b(x,y)$ switching between growths $p$ and $q$.

The authors' stated motivation is that these two research lines—double phase problems in the sense of Zhikov (2603.01100) and mixed local–nonlocal operators of the form $-\Delta_p u - (-\Delta)_q^s u$—have developed largely independently. Existing double phase theory treats purely local or purely nonlocal operators, while mixed operator theory has been restricted to homogeneous growths. The paper claims that a genuine coupling of local and nonlocal effects inside a double phase structure had not previously been investigated, and positions itself as filling this gap. The mixed character of such operators has physical motivation through superpositions of Brownian motion and Lévy flights arising in optimal foraging models.

Functional setting

Since both models involve double phase structures, the variational analysis requires generalized Musielak–Orlicz Sobolev spaces. For the first problem, the authors introduce the space

$$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$

where the modular $\xi$ sums the fractional $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$0-Gagliardo energy over $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$1 with the local double phase energy $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$2. A key structural result (Proposition 3.x in the paper's numbering) establishes that $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$3 and the standard Musielak–Orlicz Sobolev norm $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$4 are equivalent on $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$5, so that $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$6 coincides with the set of functions in $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$7 vanishing outside $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$8. Consequently $$\operatorname{div}(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u)$$9 is reflexive and embeds continuously into $p^* = Np/(N-p)$0 for $p^* = Np/(N-p)$1 and compactly for $p^* = Np/(N-p)$2.

For the second problem, since neither $p^* = Np/(N-p)$3 nor the fractional Musielak space $p^* = Np/(N-p)$4 embeds into the other, the natural energy space is the intersection

$p^* = Np/(N-p)$5

with norm $p^* = Np/(N-p)$6, where $p^* = Np/(N-p)$7 satisfies the $p^* = Np/(N-p)$8-condition. The compact embedding $p^* = Np/(N-p)$9 for $p$0 underpins all compactness arguments. The paper also verifies norm–modular relations analogous to those of Liu–Dai for both spaces.

Fractional $p$1-Laplacian with local double phase: existence and multiplicity

For the concave–convex problem with exponents $p$2, the energy functional $p$3 is treated on the Nehari manifold $p$4, decomposed via fibering maps into $p$5 and $p$6. Three lemmas carry the argument: coercivity on $p$7; emptiness of $p$8 for $p$9, an explicit threshold involving $b(x,y)$0, $b(x,y)$1, and Sobolev constants; and a fibering-map analysis showing each ray meets both $b(x,y)$2 and $b(x,y)$3. An implicit function lemma provides a local parametrization of $b(x,y)$4, which is used to produce minimizing sequences that are Palais–Smale sequences.

Compactness at critical growth is handled by a level-splitting argument: any $b(x,y)$5-sequence converges strongly provided

$b(x,y)$6

where $b(x,y)$7 is the best Sobolev constant and $b(x,y)$8 is explicit. The main result asserts that for all sufficiently small $b(x,y)$9 there exists a nontrivial, nonnegative solution with negative energy, obtained as a minimizer on $p$0 after replacing a sign-changing minimizer by a suitable scaling of its absolute value.

The Brezis–Nirenberg variant ($p$1, $p$2) is handled by the same scheme with threshold $p$3, again yielding a nonnegative negative-energy solution for small $p$4. Under the additional assumption $p$5, multiplicity is obtained by genus theory applied to a truncated functional $p$6: the truncation makes the functional coercive and bounded below while agreeing with $p$7 on negative sublevels, and a genus estimate on finite-dimensional subspaces shows every symmetric sublevel set $p$8 has arbitrarily large genus. The conclusion is that for all $p$9 the critical problem admits infinitely many distinct solutions, all with negative energy. This is the strongest quantitative claim of the paper's first part; it relies on the strict negativity of the critical levels, guaranteed by the smallness condition on $q$0 relative to $q$1.

Local $q$2-Laplacian with fractional double phase: three solutions without Ambrosetti–Rabinowitz

The second model assumes a Carathéodory nonlinearity $q$3 that is superlinear at infinity with respect to the $q$4-growth ($q$5 uniformly), $q$6-sublinear at the origin, satisfies a monotonicity condition on $q$7, and is strictly increasing in $q$8 on each half-line. Notably, no Ambrosetti–Rabinowitz condition is imposed, so the standard energy-bounding device is unavailable and must be replaced by direct estimates using the monotonicity assumptions.

Two technical pillars support the results. First, the principal part operator $q$9 is shown to satisfy the $-\Delta_p u - (-\Delta)_q^s u$0-property, combining the classical Colasuonno–Pucci–Varga argument for the local term with the fractional Musielak compactness theorem of de Albuquerque–de Assis–Carvalho–Salort. Second, the truncated functionals $-\Delta_p u - (-\Delta)_q^s u$1 satisfy the Cerami condition; boundedness of Cerami sequences follows from a Fatou-lemma contradiction argument exploiting the uniform superlinear growth, and strong convergence from the $-\Delta_p u - (-\Delta)_q^s u$2-property.

With the mountain pass geometry verified (positive lower bound near zero, $-\Delta_p u - (-\Delta)_q^s u$3 along nonnegative directions), the paper obtains one nonnegative and one nonpositive nontrivial constant sign solution, shown to be genuinely signed by testing against the opposite part and using monotonicity inequalities.

The sign-changing result is more delicate. On the constraint set

$-\Delta_p u - (-\Delta)_q^s u$4

the authors prove, via the Poincaré–Miranda theorem applied to the map $-\Delta_p u - (-\Delta)_q^s u$5, that every function with nonzero positive and negative parts admits a unique pair $-\Delta_p u - (-\Delta)_q^s u$6 placing it in $-\Delta_p u - (-\Delta)_q^s u$7, and that this pair maximizes $-\Delta_p u - (-\Delta)_q^s u$8 along the two-parameter scaling. Combined with the strict increase assumption on $-\Delta_p u - (-\Delta)_q^s u$9, this yields the key inequality $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$0 with equality only at the constraint point. The infimum $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$1 is then shown to be positive and attained, using a priori bounds $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$2 on $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$3 to prevent loss of either component in the limit. Finally, a quantitative deformation lemma argument shows the minimizer is a genuine critical point: if not, the deformation would produce an element of $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$4 with energy strictly below $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$5. The outcome is a least energy sign-changing solution, so that problem (1.3) possesses at least three nontrivial solutions—one positive, one negative, one sign-changing—all obtained without the Ambrosetti–Rabinowitz condition.

Limitations and open questions

Several restrictions qualify the results. All existence thresholds are smallness conditions on $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$6; the paper does not address whether solutions persist for large $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$7, nor does it identify an extremal parameter as in some concave–convex literature it cites. The multiplicity theorem requires $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$8 and the specific Brezis–Nirenberg scaling $$W := \overline{C_c^\infty(\Omega)}^{\,\|\cdot\|_{L^\xi}},$$9; general weights are not covered. For the second model, the nonlinearity must satisfy the uniform superlinearity and the two monotonicity conditions (H2)(v)–(vi); whether the least energy sign-changing solution exists under weaker or nonuniform hypotheses remains open. Regularity of solutions, uniqueness up to sign, and nodal properties beyond existence are not studied. The paper also leaves open the qualitative question of how the interplay between the degeneracy sets of $\xi$0 and $\xi$1 affects the structure of the solution set—for instance, whether the number of sign changes can be controlled by the geometry of these sets.

Conclusion

The paper constructs the variational machinery needed to treat double phase problems with genuinely mixed local–nonlocal leading parts, introducing two tailored function spaces and verifying their reflexivity, embeddings, and modular relations. Within this framework it delivers: a nonnegative negative-energy solution for a concave–convex mixed problem with fractional $\xi$2-Laplacian and local double phase operator; infinitely many negative-energy solutions for the associated Brezis–Nirenberg problem via genus theory; and, for the converse coupling with a fractional double phase operator, three nontrivial solutions including a least energy sign-changing one, all without the Ambrosetti–Rabinowitz condition. The results extend both double phase theory and mixed operator theory to a setting where neither alone applies, and the techniques—fibering maps on Nehari manifolds, level splitting at critical growth, Poincaré–Miranda plus quantitative deformation—are transferable to further mixed double phase models such as Kirchhoff or singular variants.